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

The diffinition of number.

NVMBER IS AS much to saye, as a multitude cōpounde of many vnities, as two is compounde of twoo vnities, three is compounde of three vnities, foure of foure vnities, fiue of fiue vnities ten of ten, fourtene of fourtene, fif∣tene of fiftene, twentie of twentie vnities, &c.

And therefore an vnitie is no number, but the beginning and ori∣ginall of number, as if you do mul∣tiplie or deuide an vnitie by it selfe, it is resolued into it selfe withoute any increase. But it is in number otherwise, for there can be no num∣ber, howe great soeuer it bee, but that it may continuallye be encrea∣sed by adding euermore one vnitie vnto the same.

¶Numeration. The first Chapter.

NVmeration is the arte whereby to expresse and declare the value of any summe or numberexpand_less proposed, and is of twoo kyndes, the one gathereth the value of a summe proposed, and the other expresseth any summe cō∣ceaued by due figures & places, for the value is one thing, & the figures are another thing: & that commeth partlye by the diuersitie of figures, but chiefly of y^e places wherin they be orderly set. And first marke, that there are but ten figures or charac∣ters which are vsed in Arithmetick, wherof nine of thē are called signi∣fying figures, & the tenth is called a Ciphar, which is made like an 0, & of it self signifieth nothing, but it be¦ing ioined w^t any of y^e other figures, encreseth their valu, & these be thei.

1	2	3	4	5	6	7	8	9
one	twoo	thrée	foure	fiue	sixe	seuen	eyght	nine.

Also you shall vnderstande that euerye one of these fygures hath twoo values: One is alwaye cer∣taine and hath hys signification of hys owne forme, and the other is vncertaine which hee taketh of his place.

A place is called the seate or rome that a figure standeth in,* 1.1 and howe many figures soeuer are written in one summe or numberexpand_less, so manye places hath the whole value therof. And that is called the first place (which nexte is towarde the ryghte hande) of anye summe, and so reckning by order towarde the lefte hand, so that that place is last which is nexte the lefte hande. And contrarywyse, when you expresse the value of the fygu∣res in any summe, you muste be∣gynne at the lefte hande, and so recken toward the right hand.

Euerye of these nyne fygures, (which are called signifying fygu∣res) hath his own simple value whē

hee is founde alone, or in the first place of any summe. In the seconde place towarde the left hande, he be∣tokeneth his own value ten times. As 70. is, seuen times ten: that is to saye seuentie. 80. is eyght tymes ten: that is to say eyghtie. In the third place euery figure betokeneth hys owne value a hundreth tymes. As 700. in that thirde place betoke∣neth a hundreth times 7. that is to say, seuen hundred: In the fourth place euerye figure betokeneth hys owne value a thousande times. As 7000. is seuen thousande, and 8000. is eight thousande. These foure first places must be had perfectly in minde, yea and that by heart, for by the knowledge of them you maye expresse all kinde of numbers how great so euer they bee.

In the fift place euery figure be∣tokeneth his owne value ten thou∣sand times. As 70000. is ten times seauen thousande, that is to saye

seuentie thousande: In y^e sixt place euery fygure standeth for his owne value, a hundreth thousand times. As 700000. is seuen hundreth thou∣sande. The seauenth place M. M. times, or a million: as 7000000. is seuen M. M. or seuen millions. And the eyght place ten M. M. times, or ten millions, so that euery place to∣warde the left hande, excedeth the former ten tymes. But nowe for the easy reading, and redye expres∣sing orderly of any summe propo∣sed, you shall practise thys maner, folowing. And for example I pro∣pone this number 765432658. in the which are .ix. places. In the fyrste place is .8. & betokeneth but eyght, in the seconde place is 5. and beto∣keneth ten times fiue y^t is fiftye, in the third place is 6. and betokeneth a hundreth times sixe, that is vj. C. In the fourth place is 2. and that is twoo M. And 3. in the fifte place is ten M. times 3. that is xxx. M. So .4.

in the firste place is C. thousande times 4. foure, that is foure. C.M. then .5. in the seuenth place is a M. M. times 5. y^t is fiue M.M. or rather fiue millions. And 6. in the eyght place is six times ten millions, y^t is lx. millions. And last of al .vij. in the ix. place, is vij. C. millions. Now fo∣loweth y^e practise. First put a pricke ouer the fourth figure, and so ouer the seuenth, and likewise ouer the tenth. And also ouer the 13.16. or .19. if you had so many, and so still lea∣uing twoo figures betweene euerye twoo pricks and these roomes from one prick to an other are called ter∣naries,* 1.2 then you muste pronounce euery three figures from one pricke to an other as though they were written alone from the rest. And at thende of their value, adde so many times a thousand, as your number hath pricks (that is to say, if there be but 1. prick, it is but 1. M. if 2. pricks a M.M. or else a million, if 3. prickes

M.M.M. or a M. million, & so conse∣quētli of al other figures folowing) Then come likewise to the next iij. fygures, and sound them as if they were a part from the rest, and adde to their value so many times thou∣sandes as there are pricks betwene them & the first place of your whole number. And so doe by the next iij. figures folowing & of all y^e rest like∣wise as in example. 451234678567. The first prick ouer 8. in the fourth place, which is the place of a M. the seconde pricke is ouer 4. in the se∣uenth place, which is the place of a M.M. or one million, y^e thirde prick is ouer the tenth place which is the place of a M.M.M. or of a M. milli∣on, as in the former example. Then for the expressing of thys number by the value of euery figure accor∣dinge to the place wherein they stande, you shall fyrste begynne at the last prick ouer 1. and take it and the other twoo fygures 5. and 4.

which doe folowe hym, and value them alone and they are foure Clj. MMM. or else CCCCli. M. milli∣ons. Then take the other three fy∣gures from 1. to the next pricke, and value them as if they were a parte from the other, and they are .234. which are CCxxxiiij. millions, or 234. MM. Then come to the thirde prick ouer .8. and take the other two figures behinde it, and recken them likewise as if they were alone, and they are sixe Clxxviij.M. And laste of all come to the other thre figures which remayne, that is .567. and they are fiue Clxvij. Thus y^e whole sum of these figures, is four Clj.M. two Cxxxiiij. millions, six Clxxviij. M. fiue Clxvij, as before.

* 1.3Note also that whole number is deuided into thre kindes, that is to say, diget number, article and mixt or compounde number. The diget number,* 1.4 is all maner of numbers vnder ten, which are these nine fy∣gures,

1 2 3 4 5 6 7 8 9. of the which I haue spoken before.* 1.5 The Article number is any kinde which begin∣neth with a Cipher as this 0. and they may euer be deuided iust by 10. without anye remaine, as these, 10. 20. 30. 40. 50. 100. & all other such like. The mixte or compounde number,* 1.6 conteineth diuers and many arti∣cles, or at the least one article, and a diget, as .11. 12. 16. 19. 22. 38. 108. 1007. and so forth. And as any article nū∣ber may be made a compounde, by putting therto a diget, euen so likewyse euery cōpounde number, may be made an Article number by adding ther∣vnto a 0.

¶ And here foloweth a briefe re∣hersall of the order and Deno∣minatours of the places. And this shalbe suffici∣ent for Nume∣ration.

The order of the places.	Tenth.	Ninth.	Eyght.	Seuenth.	Sixte.	Fyfthe.	Fourth.	Thyrde.	Seconde.	Fyrst.
	4	3	2	1	0	1	8	3	4	5
	M. of Millions.	C. of Millions.	X. of Millions.	Millions.	C. of Thousandes.	X. Thousandes.	Thousandes.	Hundrethes.	Tenthes.	Vnities.	The Denominatours of the places.

Addition in whole number. Chap. 2.

ADdition is as muche as to bring togither two summes or numbersexpand_less or more into one, as if there were due to any man 223. li. by some one bodye, and 334. li. by another, & 431. by another, & you would know howe many poundes is due to the same mā in all, these three summes shall you set downe orderly the one vnder the other writing y^e greatest summe highest and the next to the greatest vnder it, and the least sum vnder the last, in such sort y^t the first figure of the one summe be directly vnder the first fygure of the other, & the seconde vnder the second, and so forth in order. When you haue thus done, draw vn∣der them a straight line, & then wil they stand thus.

[illustration] example of numbers to be added, with a horizontal line below the bottom number

431

334

223

Nowe beginne alwayes at the firste places towarde your right hande, and put togither the three

first fygures of these thre summes, and looke what commeth of them, write that vnder them be∣neath the line, as in saying

431

334

223

8

3.4. and 1. being put togi∣ther doe make 8. wryte 8. vnder three as thus.

And then go to the se∣conde fygures and do like∣wyse:

431

334

223

88

as in saying. 2.3. & 3. maketh 8. write 8. vnder 2. as here you see.

And likewyse doe wyth the fy∣gures that be in the thirde place, in saying 2. 3. and 4.

431

334

223

988

are 9. put nine vnder them, & so wyll your whole sūme appere thus: whereby you maye perceaue that those thre summes being added togither doe make 988. li. And this is the art of addition according to his simpli∣citie, if the sum of any place do not exceede a diget number. But in case y^e sum of ani one place cannot be ex∣pressed

by one figure, but by twoo, you shall put y^e first of those figures vnder the line, and keepe the other in your minde, for to adde it vnto the first figure of y^e next place. And if y^e same next place cannot be aua∣lued but by two figures, you must in lyke maner put the first of those figures vnder the lyne, and reserue the seconde for the other place next after, and thus must you doe from one place to another vntil you haue come to the last place, where in case you doe finde that the summe be of twoo figures, you muste set them both downe bycause it is the ende of that worke, as in this example.

734682456

450932345

13467891

4672123

1203754815

Where the fyrst figures are 3.1.5.6. which added togither maketh 15. & for that, that 15. is of twoo figures, I doe put the fyrste figure 5. vnder the line, and keepe the second figure (which is 1.) in my minde, y^e which. I must adde with the next fygures of the seconde place, that is to saye wyth 2.9.4. and 5. the whych togi∣ther make 21. I write 1. vnder the line for the seconde fygure of that addition, that is to saye after 5. and I kéepe 2. to be added vnto the third place the which wyth the other fy∣gures 1.8.3.4. doe make 18. therfore I put 8. next after 1. in the thirde place vnder the line, and keepe 1. to bee added vnto the fygures of the fourth place, which is wyth 2.7.2.2. the which with the 1. that I keepe do make 14. I set downe 4. for y^e fourth fygure (vnder y^e line) that is to say, after 8. and I keepe 1. to bée added vnto the figures of the fift place, the which is 7.6.3.8. with the 1. y^t I kepe

maketh 25. I put 5. in the fift place vnder the lyne next after 4. and keepe 2. in minde to be added with the fygures of the sixt place, that is wyth 6.4.9.6. and that 2. whych I keepe, maketh 27. I write downe 7. vnder the line in the sixt place, and I kepe twoo which I adde with 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 y^e eyght place and they are 10. I do put 0. vn∣der the line in the eyght place, and then I adde 1. vnto the ninth place, that is to saye with 4.7. and they make 12. the which 12. I write at length vnder the lyne bicause it is the ende of thys addition, and thys is to be done of all such like. And for the easier vnderstanding of that which wee haue spoken of addition, you may examine these twoo other exāples folowing, in y^e which y^e first hath these nūbers. 3570.2763.579.28.

which beyng added togither, doe make thys number 6940, and in the seconde example doth resulte thys number 51683. by adding to∣gither of these numbers, 47630, 3756, 272, 25, as here vnder written.

The numbers to be added.	3570,	47630
	2763	3756
	579	272
	28	25
The line put between.
The summe of this addition.	6940	51683

Of Substraction in whole number. The. 3. Chapter.

SVbstraction teacheth howe you shal abate one lesser nūber from a greater, & what ther doth remain after that you shal haue abated the same, I speake not of the abating of one egall number, from an other egall vnto it, for the facilitie ther∣of requireth no rule.

In Substraction are founde three nūbers, the one is y^t, from y^e which the substractiō is made. The second is the number y^t is to be substracted and the third is the number which remaineth after y^e substractiō is en∣ded. As when I would substract. 25. frō 40. The 40. is the nūber frō the which y^e substractiō is made. 25. is y^e nūber to be substracted, & 15. is the nūber which remaineth after you haue done the substractiō, here folo∣weth y^e practise. You shall put y^e les∣ser nūber vnder the greater in such

sorte that euerye figure of the one number may aunswere vnto eue∣rye figure of the other orderly, and then draw a right line vnder those two numbers as you did in Additi∣on. Then must you beginne at the right hande and take the first figure of the vndermost number & abate that frō the firste figure of y^e vpper∣most number, & that which remai∣neth you muste set vnderneth the line right vnder y^e figure which you haue substracted. Then afterwarde take likewise the second figure of y^e nethermost nūber, and abate that also from the seconde fygure of the higher number. The thyrde from the thirde, and so forth of all the rest tyll you come to the ende, putting alwaies y^e remaine of euery figure vnder the line in hys order, exāple. I wyll substract. 2345. from 9876. after that I haue put them downe according to the maner aforesaide.

9876

2345

7531

I take firste 5. from 6. and there resteth 1. the which I set vnder the line right against 5. Secondly I a∣bate 4. from 7. and there resteth 3. the which I set in the seconde place vnder the line, next after 1. Third∣lye, I abate 3. from 8. and there re∣steth 5. The which I put vnder the line in the thirde place, fynally I do abate 2. from 9. and there resteth 7. the which I put vnder y^e line in the fourth and laste place, and thus is this Substractiō ended, by y^e which there resteth. 7531.

But when twoo figures of one likenesse doe chaunce to meete, so y^t the one must be abated from the o∣ther, as if I should abate 7. from 7. there remaineth nothing, and then must I set a cipher 0. vnder the line. But whē the fygure which is to be abated, doth exceede y^e figure which is ouer him, so that it cannot be ta∣ken out of the same figure. Then muste you abate the nether figure

from 10. And y^t which doth remaine you shall adde vnto the same figure which is vppermost. And the sum which commeth therof shal you set vnder the line. But whensoeuer you doe borrowe any such 10. of the ouer number: you must adde 1. vn∣to the next nether figure folowing which is to be abated. And there is nothing else to be done in substrac∣tion. Exaumple, I wyll substract 93576. from 4037479. after that I haue placed my twoo numbers,

4037479.

93576.

3943903.

as I ought to doe, I doe first abate. 6. frō 9. and there resteth 3. then I put the 3. vn∣der the line ryght a∣gainst 6. And secondely I abate 7. from 7. And there resteth nothing. I doe put a cipher 0. vnder the line right against 7. in the second place. Then I come to the thyrde place where I fynde 5. which I cannot bate frō the figure ouer him, which

is but 4. therfore I doe abate it frō 10. as before I taught and there re∣steth 5. the which I doe adde wyth the 4. which is ouer him, and that maketh 9. I put 9. in the third place vnder the line for the thirde figure. Fourthly, for the 10. whych I boro∣wed I adde one vnto the next ne∣ther fygure which is three, and they make 4. the which I do abate from the ouer fygure 7. and there re∣steth 3. I put 3. vnder the lyne for the fourth figure. And then I come to the fift place where I doe fynde 9. which I cannot abate from the fi∣gure ouer him which is but 3. but I abate 9. from 10. and there resteth 1. the which I doe adde wyth 3. and they make 4. I put 4. vnder the line for the fift figure. And if it wer not, for that I did last borow 10. the substractiō should haue been ended. But for bycause that I must (for euery suche ten that I borowe) al∣wayes adde 1. vnto the next nether

fygure folowing, I must therefore proceede vnto the substraction. And for y^t that there is no other fygure folowing in the nether number, it shal suffise to haue kept the vnitie, and to abate it from the next ouer fygure. But I finde there 0. & can∣not abate 1. from 0. therfore I abate it from 10. and there resteth 9. which I doe put vnder the line in the sixte place, finallye for the ten which I borowed, I keepe 1. in minde. The which I do abate from 4. and there remaineth 3. the which I do put vn∣der the line in y^e seuenth place after 9. And y^e operation is thus ended.

An other example.

576084026

485675437

90408589

But if there were many num∣bers to be substracted from one nū∣ber alone, then must you fyrst adde those numbers togither according vnto the doctrine of the Chapiter

going before, and thē to make your substraction as aboue saide. As if I woulde abate these three summes 123.234.456. from. 98925. first I doe adde the three summes into one, & they are. 813. The which I do abate from 98925. and there resteth. 98112.

Of Multiplication. Chapter. 4.

IN multiplicatiō there are iij. numbers to be noted, y^t is to say the numbre which is to be multiplied, the which wee will call the Multiplicande: and the num∣ber by the which we multiplie, wee call the multiplyer, or multiplica∣tour. And y^e thirde number is that which commeth of the multiplica∣tion of the one by the other, which is called the product. As when I would know how much mounteth 10. multiplied by 9. y^t is to say how much are ten times nine. I fynde that they are worth. 90. then 10. is y^e multiplicand, 9. is the multiplier.

And 90. is called the product. Thē for to multiplie, is none other thing, but to finde a number which cōteineth the multiplicande so ma∣ny times, as the multiplier contei∣neth vnities, As 10. multiplied by 9. doe make 90. as before said. And 90. conteineth 10. so many times, as 9. conteineth vnities, that is to saye nine tymes.

In multiplication, it forceth not much which of the twoo num∣bers bee the multiplicande, nor which bee the multiplier. For ten multiplied by 9. maketh as many as 9. multiplied by 10. yet neuerthe∣lesse it shall be more commodious that the lesser number be alwayes the multiplier.

And for that, that the multy∣plication of figures the one by the other, is y^e most chief & necessariest kind wherby to know how to work in the multiplication of compound numbers, and that euery mā hath

not at the fingers ende: I wil ther∣fore giue you here certeine easye wayes of multiplicatiō of diget nū∣bers. When you would multiplye two simple figures, or digets y^e one by the other, abate eche of those dy∣get numbers from 10. Then mul∣tiplie the two remaines the one by the other. And if the sum do exceede 10. write only the first figure, & kepe the other to be added to the next o∣peration, which is thus. Adde your two simple figures togither: And of y^t which resulteth of y^e addition, take onely the first figure, vnto the which you must add y^e vnity which you keepe before. And y^t shall be the second figure of the sum which you do seeke. Exāple, I would multipli 7. by 6. I take 7. frō 10. and there re∣steth 3. likewise I abate 6. from 10. and there resteth 4. then I say thus 3. times 4. make 12. I write 2. for my first figure, & I keepe 1. in my mind, thē I adde 6. w^t 7. & they are 13. of the

which I caste away the seconde fy∣gure 1. and I take only the fyrst fy∣gure 3. vnto the which I adde y^e vni∣tie which I kepte, & they make 4. which I write in the seconde place, after 2. And thus I finde 42. which is the valure of 7. multiplied by 6.

Otherwise, and all commeth to one effect, set down your two diget numbers the one right ouer the o∣ther, & right against euery of them towarde the right hande write hys owne distaunce from. 10: Then multiple the twoo differences to∣gither, the fygure which commeth thereof, shall you set downe vnder both the differences. But if there be two fygures, set downe but the fyrst, and keepe the other in your minde afterwardes abate (from one of y^e two diget nūbers) the dif∣ference of the other diget number that is to say, crossewise. And vnto the remaine adde the fygure which you kepte, and that shalbe the se∣conde

number, and thus you shall haue your multiplication. Exam∣ple of the lyke figures that is to say 〈 math 〉〈 math 〉 of 7. multiplied by 6. the distaunce of 7. vn∣to 10. is 3. And the dis∣taunce of 6. from 10. is 4. I set them downe croswayes as you see: And then I saye three times foure are 12. I set downe twoo and keepe one in my minde, then I abate 4. from 7. or else thre from 6. it forceth not from which of them: and there resteth alwaies 3. vnto the which I adde the vnitie which I kepte in my minde, and they are foure, which shalbe the se∣conde fygure of the multiplication. And thus I finde that 7. multiplied by 6. maketh 42. as in the other o∣peration. Thys practise hath no place where the 12. diget numbers (doe not exceede 10.) by adding them togither, and then is multi∣plication

easye ynough without a∣ny rule.

Another way to knowe the mul∣tiplication of symple numbers, is by thys table following: the vse wherof is thus.

Fyrst you shall vnderstande that the numbers from 1. and so downe∣wardes to 9. set in the left parte or hanging margine of this table doe betoke the multipliers of all simple numbers. And the elements or fy∣gures being put highest in euerye square roome drawing towarde your right hand right against eue∣ry of the multipliers, do signifie the multiplicands, vnto y^e multipliers of the hanging margin. And the lo∣wer or inferiour numbers in euery square roome, do betoken y^e product of that multiplication, whych is made in multiplying the vpper nū∣ber ouer it, with the fygure in the hāging margin, answering dyrect∣ly vnto y^e saide square: as by exāple,

The Table of mul∣tiplication by all the Di∣get numbers.

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
49	56	63
8	8	9
64	72
9	9
81

First bicause 1. doth not multi∣plye, I set in the vpper margin the figures frō 1. to 9. both in the higher and also in the inferiour rowes, for 1. in the hanging margine, multi∣plied by 1. the vpper number in the first square bringeth but 1. so lyke∣wyse 2. beyng the higher number in the seconde square of the vpper margine multiplied by 1. in the hā∣ging margin, bringeth two for the lower number in the second square of the vpper margine, for one times 1. maketh but 1. and 1. times. 2. ma∣keth 2. then 1. times 3. maketh 3. and 1. times 4. maketh 4. and so cō∣tinuing towarde the ryght hande vntill you come to the figure of 9. which is 1. times 9. maketh 9. Then after multiple twoo of the hanging margine by 2. the vpper number of the square nexte towarde the right hande and that maketh foure which is y^e product of 2. multiplied by 2. which 4. is set vnder the 2. for

2. times. 2. are. 4. and 2. times. 3. ma∣keth 6. then 2. times 4. maketh 8. and two times 5. maketh 10. and so continuing vnto 2. times 9. which maketh 18. The lyke is to be done with the thirde rowe, and so lyke∣wyse of all the resydue.

Exaumple, I woulde knowe what is the product of 9. multipli∣ed by 8. I seeke in y^e hanging mer∣gin the multiplier 8. and amongest the squares directlye against eyght drawing towarde the right hande, I seeke the multiplicand 9. in the higher rowe, and I fynde the pro∣duct right vnder 9. to be 72. Then 72. is the number which commeth of the multiplication of 9. by 8. and so is to bee vnderstande of all the reast of the table, which table must bee of all men learned by heart, or as they saye wythoute booke, whych being learned you shall the better attaine to the rest of multi∣plication.

To come nowe vnto the practise of multiplication, when you woulde multiplie two numbers the one by the other, you must set them down after the same maner as you did in addition, and in substraction. That is to saye, the fyrst fygure of the multiplier vnder the fyrst fygure of the multiplicande, the seconde vn∣der the second, and the third vnder the thirde, if there be so many, and then draw a right line vnder them, as in the other operations going before. After you shall multiply all the figures of the multiplicande by the multiplier, and set downe the figures (comming of any such mul∣tiplicatiō) vnder the line euery one in order.

Exaumple, I woulde multiply 123. by 3. that is to saye, I woulde knowe how much amounteth thre times one hundreth, twentie and three. The twoo numbers beyng placed in suche order as is before

said, you must beginne towardes y^e right hande: and say thus 3. tymes

123

3

369

3. are 9. write down 9. vnder the line, right against 3. for the first fygure: secondly by the same 3. you must multi∣plie y^e second figure 2. & they make 6. put downe 6. after the 9. vnder the lyne: Thirdly by y^e same 3. you shall multiplie the last figure 1. and they are but three, set downe 3. after 6. for the thirde & last figure. And thus is y^e worke ended: wher∣by you shall finde, that 123. beyng multiplyed by 3. maketh 369.

But when that of the multipli∣catiō of one figure by an other. The sum which commeth therof shalbe of twoo fygures, as it happeneth moste often, then shall you write downe the first fygure, and keepe the other fygure to be added vnto y^e multiplication of the next fygure.

Exaumple, Syx men haue gai∣ned (euery one of thē) 345. crownes

I would know how many crowns they had in all.

345

6

2070

Firste I multiplie 6. times 5. are 30. I write 0. vnder the line, & keepe 3. to bee added to the next multiplication: Se∣condely I saye 6. times 4. are 24. vnto the which I adde 3 which I re∣serued. And they make 27. I write 7. in the seconde place vnder y^e line, and I keepe 2. to be added to y^e next multiplication, thirdely I saye syxe times 3. are 18. vnto the which I adde the 2. which I keepe. And they make 20. y^e which I write al downe for bicause that is y^e last worke. And so I finde y^e 345. being multiplied by 6. do make 2070. Whē the mul∣tiplier is of many figures you must multiplie all the whole multipli∣cand by euery one of those figures, and write the productes euery one vnder hys owne fygure.

Example. I would know how many dayes are past from the na∣tiuitie

of Iesus Christe vntyll the yeare 1560. full complete. I haue to multiplie 1560. by 365. whych are the dayes of one whole yeare. The leape yeares not being recke∣ned, which haue euery one of them 366. dayes.

Fyrst by the fygure 5. I

1560

365

7800

multiplie al the higher fi∣gures, saying thus fyue times 0. maketh 0. I writ 0. vnder the line for the first figure, and bicause I keepe nothing for the next place, I proceede & say 5. times 6. are 30. I set 0. vnder the line for y^e second figure, & I kepe 3. to be added to y^e next multiplication, thirdlye I say 5. times 5. are 25. The which w^t 3. that I keepe are 28. I set downe 8. and keepe 2. to be added with y^e next multiplication. Then comming vnto the fourth and last fygure, I saye 5. times 1. are 5. the which w^t 2. that I reserued are 7: I put 7. for y^e last figure of thys fyrste operation

by the fygure 5. with the which fy∣gure we haue no more to doe. And therefore I cancell the same 5. with a little strike thorow it, to signifie y^t we haue finished with that figure. And forasmuch that in multiplica∣tion there is alwayes as many sim∣ple operations, as the multiplier conteineth figures. There resteth yet 2. operatōis to be made. I come then vnto the seconde operation, which is by y^e figure 6. by y^e which I must again multiplie al the figures of the multiplicande as I did by 5 & the first fygure (which shalbe pro∣duced) you must put one rank more lower then y^e figures of the opera∣tion euen now made by 5. not right vnder y^e first fygure of the multi∣plier 5. but vnder 6. y^t is to say: one place more forewarde than the fift toward the left hande: & one ranck more lower than the fyrst operati∣on: and you shall put afterwarde euerye of y^e other figures which cō∣meth

of the same multiplication in their order: thirdly you must make the multiplicatiō by y^e thirde figure & that which shal come thereof you must set in his ranck, as here vnder you se. And now we neede make no further discourse hereof, bicause y^t he which can doe the fyrst multipli∣cation by 5. may as easely doe all the others. It shall therefore suffice to set here vnder the examples.

1560

365

7800

9360

1560

365

7800

9360

4680

Now, if you wil know how much y^e operations thus placed do amoūt vnto, which in value are but one nūber: you must add those thre nū∣bers togither, but not after y^e same maner as we haue done in y^e chap∣ter of addition, the first figure of y^e first ranck w^t the first figure of the seconde ranck, & of the thirde: but

you muste adde them in the same sort as you shal finde them situated or placed: that is to say, the fyrst fi∣gure of the fyrst rancke alone by it selfe: the second of that ranck with the fyrst of the seconde rancke. The thirde of the fyrst rancke wyth the seconde figure of the second rancke and the first of the thirde ranck: and so of all other as hereafter doeth appeare.

And thus the 1560

1560

365

7800

9360

4680

569400

yeares do containe fiue hundreth sixtye & nyne thousande foure hun∣dreth dayes not coun∣ting herein the daies of the leape yeares, which are here in number 390. for then the whole sum of the dayes should be 569790.

Another example.

34560

2456

207360

172800

138240

69120

84879360

The summe of multiplication, when you woulde multiplie anye number by 10. you must onely adde one cipher vnto all the number. As 345. multiplied by 10. maketh 3450. if you wyll multiplie by 100. Adde vnto the whole nūber two ciphers. 00. if by 1000. adde 000. And to bee briefe. when the laste fygure of the multiplier is 1. and al the rest bee ci∣phers, add so many ciphers to your multiplicāde, as there shalbe found in your multiplier. But if in mul∣tiplying, the last figure were not 1. but that there were onely certaine ciphers in the beginning: & that the other were signifying fygures, and

likewise those of the multiplicand, then shall you put those cyphers apart, and multiplie the signifying figures of the one by the signifying fygures of y^e other. Then adde vn∣to the product of that multiplicati∣on, al the ciphers which you did be∣fore put a part. As if I would mul∣tiplie. 46000. by 3500. I put a part y^e three ciphers of y^e first, and the twoo ciphers of the seconde nūbers. And then I multiplie 46. by 35. & therof commeth 1610: vnto the whych I adde the 00000. and then the whole product will be. 161000000.

46

35

230

138

161000000

Of Diuision the fift Chap.

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

¶ The practise.

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

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

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

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

The diuidende	860
Diuisor.	4

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

860

4  (2

8

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

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

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

2

260 (21

4

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

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

2

860 (215

4

20

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

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

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

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

316215

45

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

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

1

316 215

45 (7

315

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

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

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

1

52 6215

45 (70

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

1

316215

45 (703

135

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

* 1.7But here is to bee noted, that if it happen that the fygure beynge laste founde whiche is put in the quotient, doe produce or bringe foorthe a grea∣ter noumbre (in multipliyng all the dyuisor by the same) then that whiche is ouer the saied diuisor: you muste then make the same figure of youre quotient (whiche you doe put downe) lesser by one: and after that you haue cancelled the firste multiplication, you must make a newe. And the same must bee so doone as often times: as (in decreasing the same) it produceth a lesser noumbre, or at the least, a noumbre egall to that whiche is ouer it. As in the laste woorke: for because that the diuisor, being multiplied by 3. bringeth foorthe. 135. whiche a∣mounteth more then. 121. the same producte must be cancelled. And like∣wyse the figure. 3. whiche I did put in the quotient, must bee chaunged into a figure of 2. Then by the saied 2. I must multiplie the diuisor. 45. and

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

13

316215

45 2

135 (703

90

And here is also to bee noted that the somme whiche remaineth must be alwayes lesser then the diuisor.* 1.8 Thē finally I remoue the diuisor to the 2. last figures towarde the right hande, and I seeke howe many times 4. is in 31. And for because I finde it 7. times, I put 7. in the quotient: by the whiche I multiplie the diuisor, and thereof commeth 315. the whiche I abate frō the hygher noumbre of the diuidend, and there remaineth nothing as here you may see.

13

516215

45 (7027

515

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

〈 math 〉〈 math 〉

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

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

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

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

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

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

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

Example of the diuisor proponed,

643. first, I wryte downe 643. and

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

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

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

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

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

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

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

(1227 547/043

¶ The summe of diuision.

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

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

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

¶The profe of Addition.

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

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

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

24567	2
5329
431
30377	2

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

¶The profe of substraction.

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

you haue done well, o∣therwyse

5463

3584

1879

5463

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

¶The profe of multiplication.

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

¶The profe of diuision.

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

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

¶Of progression the vi. Chapiter.

PRogression arithmetical,* 1.9 is a briefe & spedy assembling or adding together of diuers fi∣gures or nombres, euery one surmounting the other cōtinually by equall difference: as 1.2.3.4.5. &c. here the differēce, from the first to the secōd is but of 1. and so do al the other, euery one excede another by 1. still to thend. Like waies. Here .2.4.6.8. &c. do pro∣cede by the difference of 2. also 3.6.9. 12. &c. doe euery one differ from other by 3. and so may these noumbres con∣tinue. Infinitelie after this order, in adding vnto the thirde noumbre, the quantitie wherein the seconde dothe differ from the fyrste: Lyke wayes

addinge the same difference vnto the fowerth noumbre, also to the fyfte, and so vnto all the other. As .1.4. the difference of the seconde to the fyrst is 3. adde 3. vnto 4: and they are 7. for the thirde noumbre: Then adde 3. vn∣to 7: and thei make 10. for the fowrth noumbre, and so of all other.

Then if you will adde quickely the noumbre of any progession, you shall dooe thus, first tel howe many noum∣bres there are, and wryte their somme downe by it selfe, as in this example, 2. 5. 8. 11. 14. where the noumbres are 5. as you maye sée, therefore you must sette downe 5. in a place alone, [unspec 5] as I haue done here in the margent. Then shall you adde the first noum∣bre and the last together, whiche in this exaumple are .14. and 2. and they make .16. take halfe thereof whiche is .8. and multiplie it by the 5. whiche I nooted in the margente for the noumbre of the places, and the som∣me whiche amounteth of that multi∣plicacion,

is the iust somme of al those figures added together, as in this exā∣ple: 8. multiplied by .5. doe make .40. and that is the somme of all the figu∣res. An other example of parcels that are euen, as thus .1.2.3.4.5.6. in this exāple you must likewaies note doun the nomber of the places, as before is taught, and thā adde together the last nomber and the first. And the somme, whiche cometh of that addicion, shall you multiplie by halfe the nomber of the places, whiche before are noted, and that, whiche resulteth of the same multiplicacion, is the wholesomme of all those figures, as in this former ex∣ample, where the nomber of the pla∣ces is .6. I note the .6. a part, and then [unspec 6] I adde .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 moche amoū∣teth all those figures, added together.

Progression Geometricall is,* 1.10 when

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 fift the fowerth. &c. As .2.4.8.16.32, 64: here the proporcion is double.

Likewaies .3.9.27.81.243. are in triple proporcion.

And .2.8.32.128.512. are in propor∣cion quadruple.

That is to saie, in the firste exam∣ple, where the proporcion is double, euery nomber containeth the other .2. tymes. In the seconde example of tri∣ple proporcion, the noumbers exceade eche other thre times. And in the third example, the nombers exceade eche o∣ther fower times, and thus you se that progression Arithmeticalle, differeth from Progression Geometricalle for that, that in the Arithmeticalle.

The excesse is onelie in quantitie, but in the Geometricalle, the excesse is in proporcion.

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 nom∣ber, you must multiply the last somme in the progression. And from the pro∣ducte of the same multiplicacion, you shall abate the first nomber of the pro¦gression. And that whiche remaineth of the saied multiplicacion, you shall diuide by .1. lesse then was the nom∣ber, by the which I did multiplie. And the quocient shall shew you the sōme of all the nombers in any Progressi∣on. 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 shall abate the first nomber of the pro∣gression, 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:

which is the total somme of the nom∣bers of that progression. Like wise .4. 16.64 256.1024. whiche are in pro∣porcion quadruple: therfore multiplie 1024. by .4. and thereof cometh 4096 from the whiche abate the firste nom∣ber .4. and there resteth .4092: The whiche you must diuide by .3. and you shall finde in your quotiente .1364. whiche is the total somme of that pro∣gression, and this shalbe sufficient for progression.

¶ The .vii. chapiter treateth of the Rule of .3. called the golden Rule.

THe rule of three is the chifest the moste profitable, and the moste excellent rule of al the rules of Arithmetike. For al other rules haue nede of it, and it pas∣seth all the other, for the whiche cause it is saied, that the Philosophers did name it the golden rule. And after o∣thers opinion and iudgement, it is cal¦led

the rule of proporcions of noum∣bers. But now in these daies, by vs it is called the rule of thre, because it re∣quireth three nombers in his operaci∣on. Of the whiche three nombers, the twoo first are set in a certain proporci∣on. And in soche proporcion as their bée stablished, this rule serueth to finde out vnto the third nomber, the fourth nomber to hym proporcioned, in soche sort as the second is proporcioned vn∣to the firste. Not for that, that the fo∣wer noumbers, nor yet the three, are or bee proporcionall, or set in one pro∣porcion, but soche proporcion, as is from the firste to the seconde, ought to bee from the thirde vnto the fowerth, that is to saie, if the seconde noumber dooe conteine the firste twoo tymes or more, so many tymes shal the fowerth nomber conteine the thirde. And note well that the firste nomber, and the thirde in euery rule of three, oughte and must bee alwaies semblable, and of one condicion. And the second nom∣ber,

and the fowerth muste likewise bee of one sembleaunce 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 fo∣werth. And the seconde noumber by the thirde. The twoo multiplicacions 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 dissemble∣aunte by the other: that is to saie, the fowerth nomber by the second (which are dissembleaunte to the other twoo nombers) your twoo quocientes will bee egall.

* 1.11The stile of this rule is thus, you muste sette doune your three noum∣bers in a certaine order, as by exam∣ple here vnder shall appere. And then multiplie the thirde noumber, by the seconde. And the producte thereof you muste diuide by the firste noumber, or otherwise, diuide the firste noumber

by the seconde. And the quocient ther∣of shalbee diuisor vnto the third nom∣ber, that is to saie, the thirde noumber shall bee diuided by the quotient of the foresaied diuision, that is of the firste noumber diuided by the seconde. Or otherwise diuide the seconde noumber by the firste. And that whiche cometh into your quotiente, you shall multi∣plie it by the thirde nomber. And thus shall you haue the fowerth noumber, whiche you seke for.

¶Example.

IF .8. bee worthe .12. what are 14. worthe after the rate, or els if .8. require .12. for his pro¦porcionall, what will .14. de∣maunde? The whiche three noum∣bers 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

(for the whole producte of this multi∣plicacion) 168. the whiche (as the rule teacheth) you muste diuide by the first nomber, that is to saie by .8. and ther∣of commeth .21. And so moche 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 8/12. wher∣fore abreuie 8/12 and thei are 1/9 for your quotient, then diuide the thirde nom∣ber .14. by the saied ⅔, and you shall haue .21. as before. Or els diuide the seconde nomber .12. by the firste nom∣ber .8. thereof commeth .1.1/2. the which 1 ½ you shall multiplie by .14. and ther¦of will come .21. as is aboue saied, and thus muste you dooe of all other. And although, that the noumbers of this rule maie be founde in three differen∣ces, for sometymes thei are whole nō∣bers and broken together, sometymes broken and broken together, and som∣tymes all whole nombers, if thei bee whole nombers, you muste dooe none

otherwise, then you did in the last exā∣ple. But in case thai be broken noum∣bers, or broken and whole noumbers together, the maner and waie to dooe theim, receiueth a certaine variacion, and difficultee, accordyng to the vari∣etie of the noumbers, the whiche ope∣racion easely to doe, and vnuariablie, this rule teacheth.

The three noumbers beyng sette doune, according vnto the order of the whole noumbers aforesaied, without any broken nomber, let .1. bee put al∣waies vnderneath euery whole nom∣ber, with a line betwene them fracciō wise, as thus 8/1. and that .1. is denomi∣natour to euery soche whole nomber. When you haue whole nomber and broken, thei must bee reduced and ad∣ded with their broken nomber, and if there bée broken nomber without any whole nomber, the same broken must remain in their estate.

¶A Rule.

This beyng doen, you shall mul∣tiplie

the denominatour of the firste nomber, by the numeratour of the se∣conde, and the producte thereof againe by the numerator of the third nomber 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 cometh of this multiplica∣cion 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 clo∣thes, how many clothes wil .300. poū∣des buye me of the same price, that the twoo clothes did cost, sette doune your three nombers thus.

Lib. Clothes. Lib.

15.	2.	300.	1
		2	600
		600	155	(40
			1

And thā as you sée, multiplie the third nomber, whiche is .300. li. by .2. which is the second nōber, and therof cometh 600. the whiche .600. you must diuide by the first nōber .15. and you shal find in your quotiēt 40. whiche is .40. clo∣thes, and so many clothes shall I buye for .300. li. 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 likewise the seconde & the fowerth which you haue found are of one sem∣blaunce: but in case that the first nom∣ber 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 no∣bles will .48. poundes gaine me: Here you se that the denominaciō of the first

nomber is nobles, and the denomina∣cion of the thirde, is poundes, where∣fore, before you dooe procede to worke by the rule of three, you muste firste turne the poūdes into nobles in mul∣tiplying .48. poundes by three nobles and thei 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 deno∣minacion. Then shall you sette doune your .3. nombers in order thus.

If .12. nobles doe gaine me .6. no∣bles, what shall .144. nobles gaine? the whiche. 144. are the nobles which are in .48. li. Then multiple 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 muse, to see all the three

nombers in this rule of three, to bee of one denominacion, whiche can not o∣therwise bee dooen, if you reduce the third nomber, to the denominacion of the firste. But if you will reduce the firste nomber, to the denominacion of the thirde, that is to saie the .12. nobles into poundes, then shall the firste and the thirde nombers onely agree in one denominacion, and the fowerth nom∣ber 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 poun∣des, then will thei stande thus.

Poūdes.	Nobles.	Poūdes.
4.	6.	48.
		6.		Nobles.
		288.	288	(72
			44

There is yet a more exacte waie to woorke in this rule of three, whiche is thus. You must marke if the third and first nombers in the rule of thrée, maie bee bothe diuided by one like diuisor: the whiche after you haue diuided thē, you shall write doune eche of the quo∣cientes 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 chaun∣gyng the middle noumber, that is to saie the seconde, as thus, if .50. Crou∣nes doe buye me .44. yardes of clothe, howe many yardes shall I haue for 120: here you maie see that the thirde and the firste nombers, maie be diui∣ded by .10. whiche in the thirde nom∣ber is .12. tymes, and in the firste .5. ti∣mes. Wherefore you shall put .12. for the thirde nomber in the rule of three, in stede of 120: and 5. for the first nom∣ber in stede of .50. and let .44. remain still in the middest for the second nom∣ber, after this sorte as foloweth, and

then worke by the rule as before.

Crounes.	Yardes.	Crounes.
5.	44.	12.
	12.
	88.	3
	44	528 (105 ⅕
	528	355

Multiplie .44. by .12. and thereof commeth .528. diuide thesame .528. by 5. and you shall finde in your quocient 105. ⅗. and euē so many yardes should you haue founde, if you had wrought the rule of thre, by the first nōbers pro∣posed. There is yet certaine other va∣rieties, in woorkyng by the rule of three, but for that thei require the knowledge of fraccions, and because thei are not so easie as this first waie, whiche is common, therefore content your selues with this same, vntill you haue tasted the fraccions, the whiche by gods helpe I intende to set forth in seconde part of this boke, incontinen∣tly after that I haue firste taught you

the backer rule of three.

The backer rule of thre is so called: because it requireth a contrary woor∣kyng to that, whiche doeth the rule of three directe, whereof wee haue nowe treated. For in the directe rule of thrée the greater the thirde noumber is, so moche the greater will the fowerth be But here in this backer rule it is con∣trariwise, for the greater the third nō∣ber is, so moche lesser will the fowerth bee. Then, where as in the rule of .3. directe, the thirde noumber is multi∣plied 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 thesame by the thirde, and the nomber whiche commeth in the quociente, an∣swereth to the questiō. For soche prac∣tise commeth often tymes in vse: In soche sorte, that if you woorke the∣same by the rule of three directe (not hauyng a regarde vnto the Proposi∣cion of the question) you should then

commit an euident and open errour.

¶Example.

If 15. shillinges worth of wyne wyll serue for the ordinarye of 46. men, when the Tonne of wyne is worth 12. pounds: for how many mē will the same 15. shillings suffise whē the tonne of wyne is worth but eight poundes? It is certaine, that the lo∣wer the price is that y^e tonne of wine doth cost, and so many more persons will the said 15. shillings in wine suf∣fise. Therefore set downe your num∣bers thus, if 12. poundes suffise 46. men, how many will 8. poundes suf∣fise, you must multiply 46. by 12. and thereof commeth 552. the which 552. you shal diuide by 8, and therof com∣meth 69. and vnto 69. men wyll the saide 15. shillings worth in wine suf∣fyse, whē the tonne of wyne is worth but eyght poundes, as hereafter doth appere by practise.

Lib.	Men.	Lib.
12.	46.	8.
	12.		7
	92.		552	(69.
	46.		88
	552.

Likewise, a messenger maketh a iourney in 24 dayes, when the day is but 12. houres long: how many daies shall hee be vppon the same iourney when the day is 16. houres in length? Here you must perceaue, y^t the more houres are in a day, the fewer dayes wyll the messenger bee in going hys iourney. Therefore wryte downe your nūbers thus as here you mai se.

Houres.	Daies.	Houres.
12.	24.	16.	4
	12.		12
	48.		288
	24		166
	288.		1	(18

And then multiplie 24. dayes by 12. houres, and thereof commeth 288: diuide the same 288. by the thyrde

number 16. and you shall fynde 18. the which is 18. dayes, and in so many dayes wyll the messenger make hys iourney whē y^e day is 16. houres long.

Likewyse, when the bushell of wheate doth cost 3. shillings, the pen∣ny loafe of breade waieth 4. lib.

I demaunde what the same pennye loafe shall waye, when the bushell of wheate is worth but twoo shillings: here is to bee considered that the bet∣ter cheape the wheate is, the heauier shall the peny loafe waie, and there∣fore write down your 3. nūbers thus.

Shill.	Lib.	Shill.
3.	4.	2.
	3.		12
	12.		2	(6.

Then multiplie 4. lib. which is the seconde number, hy the fyrst number 3. and they make 12. the which 12. you shall diuide by the thirde number 2. and thereof commeth 6. li. & so much must the peny loafe of breade waye, whē y^e bushel of wheate is worth but

two shillings as may appeare.

And nowe, according to my for∣mer promise, shall follow the seconde parte of Arithmeticke, whiche teacheth the working by Fractions.

¶Here endeth the first parte of Arith∣metick.

The seconde part of Arithmetick, which trea∣teth of Fractions or bro∣ken numbers.

¶The first Chapter treateth of Fracti∣ons, or broken numbers, and the dif∣ference thereof.

BRoken number is as much as a parte or many parts of 1. wher∣of there are two num∣bers w^t a line betwene them both: y^t is to say, y^e one which is aboue y^e line is called y^e numerator. And y^e other vnderneth y^e line is cal∣led the denominator, as by exam∣ple, thre quarters, which must be set downe thus, ¾: whereof 3. which is the higher number aboue the line is called the numerator, and 4. which is vnder y^e line is called y^e denominator. And it is alwaies conuenient that y^e numerator be lesse in number, than the denominator. For if the nume∣rator,

and y^e denominator were egall in value: then shoulde they represent a whole number thus, as 1/1, 2/2, 3/3, which are whole numbers: by reasō that the numerators of these, and all such like, may be diuided by their de∣nominators, and theire quotientes will alwayes be but 1. But in case that the numerator doe exceede hys denominator, then it is more thā one whole: as 20/18, is more than a whole number by 2/18, other diffinition doth not hereunto appertaine. Further∣more it is to be vnderstande that the middest of all broken numbers is the iust half of 1. whole, as 6/12, 7/14, 8/16, 9/18, and other lyke, are the halfes of one whole number, wherof doth growe, and come forth 2. progressions natu∣rall: the one progreding by augmen∣ting, or encreasing, as these.

½ ⅔ ¾ ⅘ ⅚ 6/7 ⅞ 8/9 9./10. &c.

And they doe proceede infinitely and

wyll neuer reache to make a whole number thus 1/1. And the other pro∣gression, doth progrede by dimini∣shing or decreasyng, as thus.

½ ⅓ ¼ ⅕ ⅙ 1/7 ⅛ 1/9 1/10. &c.

And these do proceede infinitely, and shall neuer come to make a 0. which signifieth nothing, but shall euer re∣taine some certaine number whatso∣euer, whereby it doth appeare that broken numbers are infinite.

¶The seconde Chapter treateth of the reducing or bringing to∣gither, of 2. numbers, or many broken dissembling, vnto one broken sembling.

REduction, is as much as to bring togither, or to put in sēblaūce 2. or ma∣nye nūbers dissembling one from the other, in reducynge

them vnto a common denominator. For bicause the diuersitie and diffe∣rence of the broken numbers, doe come of the denominators part, or of diuers denominators, and for the vn∣derstanding hereof, there is a general rule whose operation is thus. Mul∣tiplie the Denominators the one by the other, and so you shal haue a new denominator common to all, y^e which denominator diuide by the perticuler denominators, and multiplye euery quotiēt by his numerator and so you shall haue newe numerators, for the numbers which you woulde reduce, as appeareth by thys example follo∣wyng.

¶Reduction in common de∣nomination.

[unspec I] IF you wyll reduce ⅔ and ⅘ togi∣ther, you must fyrst multiplie y^e two denominators the one by y^e other, that is to say 3. by 5. maketh 15. which is your commō denominator,

that set vnder the crosse, then diuide 15. by the denomi∣nator 3. & you shall haue 5. which mul∣tiply by the nume∣rator 2. and you shall finde 10. set that ouer the ⅔ and they are 10/11, for the ⅔. Afterwardes diuide 15. by the deno∣minator 5. and therof commeth 3. the which multiply by the numerator 4. and you shall finde 12. which set ouer the head of the ⅘ and they make 12/15 for the ⅘: as appeareth more plainer in the margent. 〈 math 〉〈 math 〉

2. If you will reduce ½, ⅔, ¾, ⅚, to∣gither, you must multiplie all the de∣nominators the one by y^e other, that is to say, 2. by 3. maketh 6, then 6. by 4. and mounteth 24. Last of all 24. by 6. and thereof commeth 144. for the common denominator. Then, for the fyrst diuide 144. by the denomi∣nator 2. and thereof commeth 72. the which multiplye by the numerator 1.

and it is still 72. set y^t ouer the ½ and it is 72/144, for the ½: Then diuide 144. by the seconde denominator 3. & ther∣of commeth 48: the which multiplie by the seconde numerator 2. and they are 96. which set ouer the ⅔ and they make 96/144, for the ⅔: Thē diuide 144. by the thirde denominator 4. & therof commeth 36. the which multiplye by the thyrde numerator 3. & they make 108. which set ouer the ¾ and they are 108/144 for the ¾.

Finally diuide 144. by the last de∣nominator 6, & thereof commeth 24: The which multiply by y^e last nume∣rator 5. & therof commeth 120. Which set ouer the ⅚ and they are 120/144, for the ⅚, as appeareth here by prac∣tise.

¶The exaumple.

〈 math 〉〈 math 〉

¶Reduction of broken num∣bers of broken.

IF you wyl reduce y^e broken of bro∣kē togither, as thus, the ⅔ of 7/4 of ⅘, [unspec 3] you must multiplye the numerators the one by the other to make one bro∣ken number of the thre broken num∣bers, that is to saye 2. by 1. maketh 2. and then 2. by 4. maketh 8. which is your numerator. Then

	8
⅔	¼	⅘
	60.

multiplye the Denomi∣nators the one by the o∣ther, that is to saye 3. by 4. maketh 12. and then

12. by 5. maketh 60. for your denomi∣nator, set 8. ouer 60. w^t a line betwene them, and they be 1/60 which being ab∣breuied are 2/15 and so much are y^e ⅔ of ⅙ of ⅘ as appereth in the margent.

¶Another example of the same reduction and of the se∣conde reduction.

[unspec 4] IF you will reduce ⅔ of, ¼, of ⅘, y^e ¾, of 5/7: And the ½, of the ½, of the ⅔ of ⅓. First it behoueth you of euery par∣tie of y^e brokē nūbers to make of eche of thē one broken, as by the third re∣ductiō is taught: y^t is to say, in multi∣plying y^e numerators by numerators & denominators by denominators: first, for y^e fyrst part which is ⅔ of ¼ of ⅘, you must as is said before, multi∣plie 2. by 1. and then by 4. & you shall haue 8. for the numerator lykewise multiplie 3. by 4. and the product by 5. and you shall haue 60. for the deno∣minator, so they make, 1/60 which be∣ing

abreuied are 2/15 for y^e first part, that is to say, for the ⅔ of ¼ of ⅘, se∣condely for the ¾ of 5/7 multiplye lyke∣wise the numerator 3. by 5. maketh 15. for y^e numerator, & multiplie 4. by 7. maketh 28. for the denominator, and then they be 15/29 for y^e seconde part that is to say for the ¾ of 5/7. Thirdely for the ½ of ½ of ⅔ of ⅓ multiply the nu∣merators the one by the other, that is to say, 1. by 1. and then by 2. and last by 1. and al maketh but 2. for y^e nume∣rator, likewyse multiplie 2. by 2. ma∣keth 4. and 4. by 3. maketh 12. and then 12. by 3. maketh 36. for denomi∣nator, and they are 2/36, which beyng abreuied maketh 1/18, for the thyrde part, y^e is to say for ½ of the ½ of ⅔ of ⅓. Last of all take the 2/15 the 15/28 and y^e 1/18 & reduce them according to the order of the seconde reduction, and you shall finde 1008/7560 for the 2/15. And 4050/7500 for y^e 15/28. And 420/7560 for y^e 1/18: and thus are bro∣ken numbers of broken, reduced as appereth by practise.

8	15	2
⅔¼⅘	¾ 5/7	½½⅔⅓
1008	4050	420
60	28	36	2/15	15/28	1/18
				760

¶Reduction of broken numbers, and the partes of bro∣ken togither.

[unspec 5] IF you wyll reduce ⅓ and y^e ½ of ⅓ to∣gither to bring them into one bro∣ken number, you must first set down the ⅓ and ½ as appereth in the marget 〈 math 〉〈 math 〉 with a crosse betweene them, & then multiplie the twoo deno∣minators the one by the other, that is to say, 2. by 3. maketh 6. set that vnder the crosse, then multiplie the fyrst Numerator, one by the laste denominator twoo, and that maketh 2. vnto the whych adde the laste numerator one, and they be three, whych set aboue your

crosse, so you shall finde that the ⅓ and the ½ of ⅓ doe make ⅓ which being ab∣breuied dothe make ⅓, which is as much as the ⅓ and the ½ of ⅓. Likewise if you wyll reduce the ⅔ and y^e ¼ of ⅓, you must do as before, set downe the ⅔ and ¼ with a crosse betweene them, multiplie the two denominators the one by y^e other, that is to say, 3. by 4. maketh 12. which set vnder y^e crosse as you see in the margent 〈 math 〉〈 math 〉 and thē multiply the first numerator 2. by y^e last denominator 4. and thereof com∣meth 8. whereunto adde the last nu∣merator 1. and that maketh 9. which set ouer the Crosse, so shall you finde that the ⅖ and the ¼ of ⅓ are worth 9/12, which abbreuied doe make ¼, as appeareth by example in the margent.

Reduction of whole numbers and broken togither into a Fraction.

[unspec 6] IF you will reduce whole number with broken, you muste bring the whole number into broken, as by this exāple may appere: reduce 17. ⅝ into a broken number, first you must multiplye the whole number 17. by the denominator of the brokē, which is eight in saying eight times 17. doe make 136. vnto the which you must adde the numerator of ⅝ which is 5. and all amounteth to 141. which set ouer 8. with a line betweene them, & thei wil be 141/8 so much is 17. ⅛ worth in a Fraction as appeareth here by practise.

17.		141.
8.		5.
136.	17.	8.	maketh	141./8.
5.
141.

In case you haue whole number and broken to bee reduced, with broken you muste bring the whole number into his broken, in multiplying it by the denominator of the broken num∣ber going therwith, and adde there∣unto the numerator of the saide bro∣ken number, as in the laste example, and then reduce that brokē number wyth the other broken, as here appe∣reth by this example. Reduce 10. ⅔ & 4/7 togither, first bring 10 ⅔ al into thirds, as by the syxt reduction, and you shal finde 32/3, then reduce the 32/3 and 4/7 togi∣ther, by the fyrste reduction, and you shall finde 224/21 for the 32/3: and 12/21 for 4/7, as appereth here by practise.

〈 math 〉〈 math 〉

Also in case you haue in both partes of your Reduction, as well whole number as brokē, you must alwayes put the whole into their broken (as by the syxte reduction) of either part.

¶Example.

If you wyll reduce 12. ¼ with 14. ⅔ to bring them into one denomina∣tion, fyrst bring the 12. ¼ all into four∣thes, and you shall fynde 49/4: then likewise reduce 14. ⅔ all into thirdes, and you shall haue 44/3, for the 14. ⅔, then reduce 49/4 and 44/3 togither, by the order of the fyrst Reduction, and you shall fynde 147/12 for the 49/4. And 170/12 for y^e 14. ⅔ as here by practise doth plainly appere.

〈 math 〉〈 math 〉

¶The thirde Chapter treateth of abbreui∣ation of one great broken number into a lesser broken.

ABbreuiation is asmuch as to set downe, or to write a brokē nūber by figures of lesse sig∣nification, & not diminishing y^e value thereof. The which to doe, there is a rule whose operation is thus, diuide y^e numerator and likewise the deno∣minator, by one whole number the greatest y^t you may fynde in the same broken number, & of the quotient of that numerator, make it the nume∣rator, and likewise of that of the de∣nominator make it your denomina∣tor, as by example.

1. If you wyl abreuiat 54/81, you shal vnderstande that the greatest whole number that you maye take, by the which you maye diuide the numera∣tor & denominator is 27, which is y^e half of y^e numerator, & that is a whole number, for you cannot take a whole

number out of the denominator, 81. but that there will bee either more or lesse than a whole number, therefore if you diuide 54.

54./81.
1
27
27	(2
		⅔
2
81
27	(3

by 27. you shall finde 2. for the nu∣merator, likewise if you diuide 81. by 27. you shall finde 3. for the denomi∣nator, then put 2. ouer the 3. with a line betwene thē, and you shal find ⅔ and thus by this rule the 54/81 are a∣breuied vnto ⅔, as appeareth in the margent, and so is to be vnderstande all other.

¶The forme & maner how to finde out the greater number, by the which you mai wholy diuide y^e nu∣merator & denominator (to thende y^t you may abreuiat them) is thus.

Fyrst, diuide the denominator by hys numerator, and if anye number doe remaine, let your diuisor be diui∣ded by the same number, and so you must continue vntyll you haue so di∣uided y^t there may nothing remaine, then is it to be vnderstande, that your last diuisor (wherat you did ende, and that 0. did remaine after your last di∣uision) is the greatest number, by the which 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 be abreuied. Example, of 54/11 diuide 81. (which is the denominator) by 54. which is his numerator, and there resteth 27. then diuide 54. by 27. and there remaineth nothing, wherefore your last diuisor 27. is the number, by the which you must abreuiat 54/81 as in the last exam∣ple is specifyed.

¶Another stile of abbreuiation.

2. Mediate the numerator, and al∣so the denominator of your fraction in case the numbers be euen, that is to saye, take alwayes the halfe of the numerator, and likewise of the deno∣minator, and of y^e mediation or halfe of the numerator, make it your nu∣merator, also of ½ the denominator, make your denominator, and so con∣tinue as often as you may in takyng alwayes the ½ of the numerator, and semblablie of the denominator, or else see if you may abbreuiate y^e num∣bers which doe remaine, by 3. by 4. by 5.6.7.8.9. or by 10: for you must ab∣breuiate them as often as you can by any of the saide numbers, and it is to bee noted, that with whatsoeuer nū∣ber of these, you doe abbreuiate the Numerator of your Fraction, by the same you must abbreuiate lykewyse the Denominator, so continuyng vntill they can no more bee abbre∣uied. And it is to bee vnderstande that if the Numerator and the De∣nominator

be euen numbers, as you may knowe when the fyrst fygure is an euen number, or a 0, maye you perceaue if both the Numerator and the Denominator may be abbreuied by 10. by 8. by 4. or by 2. although that some times they maye bee abbreuied by thre. And if thei be odde numbers, then must you consider if they maye bee abbreuied by 9. by 7. by 5. or by 3: but when the fyrst number, as well of the Numerator, as of the Deno∣minator are euen numbers, then may you wel knowe that such num∣bers maye bee abbreuied by 2. as is aforesaide. And if you adde the fy∣gures of the Numerator togither, in such manner as you doe in ma∣kyng the proofe by nine in whole numbers: that is, if you fynde 9. it appeareth that you maye abbreuye that number by 9. And lykewise by 3. and sometimes by 6. if you fynde 6. it maye bee abbreuied by 6. and

alwaies by 3. if you finde 3. it is a signe that you may abreuiate by 3. And by whatsoeuer number that you doe a∣breuiate the numerator, by the same must you abreuiate likewyse the de∣nominator, and if the fyrst fygures of the same number be .5. or 0. you may abreuiate them by 5. but if the fyrste fygures be both 0. they may be abre∣uied by 10. in cutting awaye the twoo Cyphers thus, as

⅔

0/0

which maketh ⅔, as sometimes by 100. thus, as

½

00/00

in cutting away the foure ciphers af∣ter this sort,

½

00/00

and then the 100/200 doe make ½, and after this maner haue I set here diuers examples, although that all numbers cannot be abreuied by this rule, that is to saye, all those which may bee well abreuyed by the fyrste rule afore∣saide.

Abreuiated.

by	10.	3840/7080	by	9.	1890/4725
by	8.	384/768	by	7.	210/525
by	6.	48/69	by	5.	30/75
by	4.	8/16	by	3.	6/15
by	2.	2/4			2/6
		½

3. Furthermore you shall vnder∣stande that sometimes it happeneth, that all the fygures of the numera∣tor are egall vnto them of the deno∣minator, which whē it so happeneth, you may thē take one of them of the numerator, and also one of them of the denominator, and it shall bee a∣breuyed as 555/888, being abreuiated af∣ter this maner cōmeth to ⅝. And yet it happeneth sometimes, that two, or many fygures of the numerator are proportioned vnto two, or many fy∣gures

of their denominators and the other fygures of the same number doe beholde the one the other in thys proportion? Then may you take two or many figures, as well of the nu∣merator as of the denominator, and by this maner the same number shal be abbreuied, as 4747/5959 which being abbreuied by this rule, do come to 47/59.

4. Also it happeneth sometimes y^t you woulde abbreuiate one number vnto the semblaunce or likenesse of another. And for to know if the same may be abbreuied, and also by what number it maye bee abbreuied, you must diuide the numerator of y^e one number by the numerator of the o∣ther, and likewise the denominator of the one by the denominator of the other, for in case that after euery di∣uision there doe remaine 0. and that the twoo quotients be egall, then is one of them the number by y^e which the saide fraction must be abbreuied, as by exāple of 115/207. I would know if

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 both the quoti∣ents 23. by the whych it appeareth y^t this number may be abbreuied by 23.

5/9.	115/207	115	••
		55 (23	207
			99 (23.

¶The 4. Chapter treateth of the assem∣bling of two or many broken num∣bers togither, as by example.

FOr to adde broken numbers togither, there is a generall rule, which is thus, if y^e nū∣bers be vnlike the one to y^e other you must reduce thē into a cōmon deno∣minatiō, which after you haue redu∣ced thē, you must then adde both the numerators togither, & set y^e product of the saide addition ouer the crosse, & diuide the same by the cōmon deno¦minator as by this exāple folowing.

1. If you wyll adde ⅔ with ¾, you must fyrst reduce the twoo fractions both into one denomination, accor∣ding to the introduction of the fyrste reduction, that is to saye, in mul∣tiplying the denominator of the firste fraction which is 3, by the denomi∣nator of the other fraction whiche is foure, and they make 12. for your common de∣nominator, y^e which 12. set vnder the crosse, thē multiplie y^e fyrst nume∣rator 2. by the last deno∣minator 4. and thereof commeth 8. which set o∣uer the ⅔, and then mul∣tiply y^e last numerator 3. by the fyrst denominator 3. and ther∣of commeth 9. which you must set o∣uer 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 which is the addition of the ⅔ wyth ¾.〈 math 〉〈 math 〉 And by∣cause

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 5/12, and so much are y^e ⅔ added with ¾ as doth appere.

¶Addition in broken numbers.

2. Also if you wyll adde ½, ⅔, ¾, ⅘, togither, you muste fyrste adde the ½ and ⅔ togither, according to the doc∣trine of the laste rule, and you shall finde 7/6: then adde ¾ and ⅘ togither by the saide last chapter, and they make 31/20. Then finally adde the 7/6 (which came of the ½ and ⅔ added togither) with 13/20, & you shall fynde by y^e fore∣saide addition that they amount vn∣to 326/120. Wherefore diuide 326. by 120. & therof cōmeth 2. and 86. remaineth, which is 86/120 of one whole, & thei being abreuied do make 43/60: & thus y^e ½, ⅔, ¾, ⅘, added togither doe amount to 2.43/00,

as here vnder doth appere.

〈 math 〉〈 math 〉

¶Addition of broken numbers of broken.

3. Furthermore, if you will adde the broken numbers of broken togi∣ther, as to adde the ⅔ of ¾ of ⅘ with the ⅚ of ½ of ⅝: first you must reduce y^e nū∣bers according to y^e order of y^e fourth reduction, in multiplying the nume∣rator of the fyrst 3. fractions, the one by the other, and of the product make your numerator, & likewise you must multiplie y^e denominators of y^e fore∣said thre fractions, y^e one by the other

and of the product make your deno∣minator, and you shal finde 24/60 for the fyrst thre broken numbers, which be∣ing abbreuied do make ⅖, thē reduce the other 3. fractiōs, by y^e saide fourth reduction, in multiplying the nume∣rators by numerators, & denomina∣tors, by denominators, as you did by the first 3. broken numbers, & you shal finde 25/96 thē must you adde y^t ⅖ which came of the fyrst 3. broken numbers, & 25/96 which are of the last 3. fractions, both togither, by the instruction of y^e first additiō & you shal find 317/480 which cannot be abbreuied, but is y^e product of y^e additiō: so much are the ⅔ of ¾ of ⅘ added with the ⅚ of ½ of ⅝ as hereaf∣ter by practise doth euidently appere.

〈 math 〉〈 math 〉

¶Addition of broken number with the partes of broken togither.

4. Likewise if you will adde the ⅓, and the ½ of ⅓ with the ⅘ and ¼ of ⅕, you must reduce the ⅔ ½ by the fyft reduc∣tion and therof cōmeth ⅚ for the ⅔ & ½, of one of the saide thirdes, thē reduce the ⅘ and ¼ by the saide fift reduction, and thereof commeth 17/20.

Last of all adde the ⅚ and 17/20 togither according to the fyrst rule of additiō, and you shall fynde 202/120 which beyng diuided bringeth 1. & 82/120 part remai∣ning, which abreuied maketh 41/60 and thus you doe perceaue that the ⅔ & ½ added with the ⅘ and ¼ doe amount vnto 1.41/60 as hereafter by practise doth plainely appere.

〈 math 〉〈 math 〉

〈 math 〉〈 math 〉

¶Addition of whole number and broken togither.

5. Also if you will adde 12. ⅘ with 20. ⅚, you may (if you will) adde 12. & 20. togither, and they make 32. & then adde the twoo broken numbers togi∣ther, that is to saye ⅘ and ⅚ by y^e order of the fyrst addition & they make 49/30: therefore diuide 49. by 30. and thereof commeth 1. and 19/30 partes remaine, which 1. you must adde vnto the 32. & the whole addition wil be 33. 19/30, or o∣therwise, you may reduce 12. ⅘ into the likenesse of a Fraction by the syxt reduction and they will be 64/5, & like∣wise by the same reductiō, reduce 20. ⅚ and they bee 125/6, then adde 84/5 with the 125/6, by the fyrst addition and you shall fynde 1009/30. Therefore diuide 1009. by 30. and therof commeth 33.19/30

as before, & as by practise of the same both the wayes, doth here vnder ap∣pere.

〈 math 〉〈 math 〉

¶The fift Chapter treateth of Substraction in broken numbers.

IF you wyll substract ⅔ from ¾ you must fyrst reduce bothe the fractions into a common deno∣mination by the fyrst reduction, and you shall fynde 8/12 for the ⅔, and 9/12 for the ¾. Therefore abate the numera∣tor 8. from the numerator 9. & there remaineth 1/12 as maye appere here by practise.

〈 math 〉〈 math 〉

2. But if you haue a broken num∣ber to be substracted from a whole number, you must borowe one of the whole nūber, & resolue it into a frac∣tion of like denomination as is that fraction, which you woulde abate frō the same whole nūber, & then abate y^e saide fraction therefrom, & you shall finde what doth remaine, as by thys example. If you abate ⅘ from 8. you must borow one of the said 8. and re∣solue it into fiftes like vnto y^e fractiō, bicause it is 4. fifts, and that 1. wil be 5. fiftes thus 5/5, therfore abate ⅘ from 5/5 & there will remain ⅕, and substract y^e 1. which you borowed from 8. and there doth remaine 7. and also the ⅕. Thus the ⅘ being substracted from 8. doth leaue 7. ⅓ as by practise dothe

plainely appeare.

〈 math 〉〈 math 〉

3. If you will substract broken nū∣ber from whole number and broken being togither: thus, as if you would substract ¾ from 6. ⅚, you may by the fyrst substraction, abate ¾ from ⅚, and there wil remaine 1/12, and the 6. doth still remaine whole, bicause y^e ¾ may bee abated from the ⅚, thus ¾ being abated from 6 ⅚ leaueth 6.1/12 as appe∣reth by practise.

〈 math 〉〈 math 〉

Lykewise if you wyll abate ⅔,

from 14. ⅖, you must fyrst reduce 14. ⅖ all into fiftes by the 6. reduction, and they be 72/5, then reduce ⅔ into a common denomination, by the fyrste reduction, and you shall fynde 10/15 for the ⅔: and 216/15 for the 72/5: thē substract the numerator 10. of the fyrst fraction, from the numerator 216. of y^e seconde fractiō, & there remaineth 206/15. Ther∣fore diuide 206. by 15. and therof com∣meth 13.11/15, and so much remaine of this substraction, as may appere.

〈 math 〉〈 math 〉

4. If you will substract whole nū∣ber and brokē from whole & broken, as thus, if you will substract 9. ¼ frō 20. ½ you muste reduce 9. ¼ into four∣thes, & likewise y^e 20. ½ into halfes by y^e syxt reduction: & you shal find 37/4 for y^e 9. ¼ And 41/2. for y^e 20. ½. Thē reduce 37/4. and 41/2 into one denomination, ac∣cording vnto the fyrste reduction and you shall fynde 74/8 for the 37/4, and 164/8 for the 41/2 thē abate the numerator of 74/8 which is 74. from the numerator of 164/8 and there remaineth 90/8 then di∣uide 90. by 8. and thereof commeth 11. ¼ which is the remaine of thys sub∣straction.

〈 math 〉〈 math 〉

¶Substraction of broken numbers of broken.

5. If you will substract, the 2/1 of ⅔ of ⅗ frō the ⅚ of ¾ of ⅞ you must first bring the ½ of ⅔ of ⅗ into one fraction by y^e 3. reduction, & the 3/6 of ¾ of ⅞ likewise in∣to one fraction by y^e same reduction, & you shall fynde 6/30 for the fyrst 3. brokē numbers, which being abbreuied do make ⅕: & for the other 3. brokē num∣bers, you shal finde 105/192: which being likewise abbreuied do make 35/64, then you shal substract ⅕ frō 35/64 by y^e instruc¦tiō of y^e first substractiō, in reducing both y^e fractiōs into a cōmon denomi∣nation, as before is done, & you shall find remaining 111/320 as may appere by example.

〈 math 〉〈 math 〉

¶The sixt Chapter is of multi∣plication in broken numbers.

FIrst for to multiplie in brokē nūber, there is a rule which is thus, multiply the nume∣rator of the one fractiō by the nume∣rator of the other. And then diuide y^e fraction if you may, or else abbreuiate it, and you haue done, but if there be whole number & brokē togither, you muste reduce the whole nūbers into broken, & adde thereunto the nume∣rator of his broken, and thē multiply as is before saide, as also hereafter by examples shall more plainely appere.

1. If you wyll multiply ⅔ by ¾, you must multiplie the numerator 2. by y^e numerator 3. & therof commeth 6. for the numerator: Likewise multiplye the denominators the one by y^e other, y^t is to say 3. by 4. & thereof commeth 12. fpr the denominator, so that thys multiplication commeth to 6/12, which

being abbreuiated doe make ½ and so much amounteth the multiplication of the ⅔ by ¾ as by practise.

	6
⅔		¾
	12.

2. Likewise if you will multiplye a broken number by whole number, or whole number by broken, which is all one, as ⅘ by 18, or else 18. by ⅘, you must set 1. vnder 18. thus 8/1: and thē multiply 18. by the numerator 4. and thereof commeth 72. y^e which di∣uide by the denominator 5. & thereof commeth 14. ⅖ for the whole multi∣plication, or otherwise abate from 18. his ⅕ part which is 3. ⅗, and there re∣maineth 14. ⅖ as aboue.

	72		22
⅘		18/1	72
	5		55	(14 2/5

	18		3
1••/1		⅕	18	18
	5		5 (3.	3	⅗
				14	⅖

3. Also if you wil multiply a whole number, by whole number and bro∣ken, or else whole number & broken by a whole number, which is al one. As by example, if you multiplye 15. by 16. ¾ or else 16. ¾ by 15. First reduce 16 ¾ all into fourthes, in multiplying 16. by the denominator of ¾ whych is 4. and therof commeth 64. wherun∣to adde the numerator 3. and it ma∣keth 67/4 which multiplie by 15/1 accor∣ding vnto the instruction of the laste example, & you shall finde y^e product of this multiplication to be 251. ¼ as by practise doth here appere.

					67
	67		1005		15
16	¾	1••/••		••7/4	335
			4		67
					1005

2 1
1005
444 (251.	¼

4. And if you wyll multiplye a broken number, by whole number and broken, or else whole number & broken by a broken. As by example, if you wil multiplie ¼ by 18. ⅔, or else 18. ⅔ by ¼, which is all one: you must reduce the whole number into hys broken by the syxt reduction. And you shall fynde 56/3, which you shal multi∣plie by the ¼, after the doctrine of the first multiplication, that is to say: in multiplying the Numerator 56. by y^e Numerator of ¼, which is 1. And it is still 56. bicause 1. doth neither mul∣tiplie nor diuide. And likewise you must multiplye the Denominator 3. by the Denominator 4. and it ma∣keth 12. thē diuide 56. by 12. and there∣of commeth 4. ⅔. And so muche

amounteth the multiplication of the 18. ⅔ multiplyed by ¼ as by example.

	56		56		18
18	⅔	¼		••6/3	56
			12		12 (4 2/3

5. If you wyll multiplye whole number and broken, with whole and broken, you muste fyrste put eyther whole number into hys broken, ac∣cording to the instruction of the syxte reduction, and then multiply the one numerator by the other, and of the product make your numerator. And likewise multiply the denominators the one by the other, and thereof make the denominator, then diuide the numerator by the denominator, and the quotient shal be the encrease of thys multiplication. Example, If you woulde multiplye 12. ⅘ by 6. ¾: fyrst by the syxt reduction the 12. ⅘ wil make 64/3, and the 6. ¾ wyll make 27/4, then multiplye the numerator 64.

by the numerator 27. and therof com∣meth 1728. for the numerator. And then you must multiply the denomi∣nator 25. by the denominator 4. and they doe make 20. then diuide 1728. by 20. and thereof commeth 86. ⅖ for the whole multiplication, as by ex∣ample.

		1728		64
	64		27	27
12	⅘		6 ¾	448
		20		128
				1728
		1
		1728
		200 (86.
		2

6. If you wyll multiplie one bro∣ken number by many broken num∣bers, thus: As to multiply ⅔ by 5/7 & by 4/9: you must multiply y^e numerators of al the fractiōs, y^e one by the other, & of the product make y^e numerator y^t

is to saye: 2. by 5. and they be 10. then 10. by 4. and they be 40. for the Nu∣merator. Likewise you muste multi∣ply the denominators the one by the other, that is to say 3. by 7. maketh 21. then 21. by 9. maketh 189. for the de∣nominator: then set 40. ouer the 189 wyth a line betweene them, and they make 40/189. And so much amounteth the whole multiplicatiō of the ⅔ mul∣tiplied by 5/7 and 4/9 as by example folo∣wing. And thus is to be vnderstand of all such lyke.

			2	3
			5	7
	40		10	21
⅔	5/7	4/9	4	9	40/189
	189		40	189

¶The 7. Chapter treateth of Diuision in broken numbers.

NOte that in Diuision of brokē numbers, you must sette your diuisor downe fyrst, next vnto the lefte

hande, and the diuidende or number which is to bee diuided alwaies to∣warde the right hande. And then multiplie crosse wise, that is to saye the numerator of your diuisor by the denominator of the diuidende, and the product shalbe the denominator, which afterwarde shall bee your Di∣uisor. And lykewise you must multi∣plie the Denominator of your fyrste number, that is to saye of your Diui∣sor: by the Numerator of the Diui∣dende, which afterwarde shall be the Diuidend, and that must be set ouer the Crosse, and the Denominator vn¦der the Crosse, then shall you diuide the Numerator by the Denomina∣tor if it maye bee diuided, if not, you must abbreuiate them, as hereafter by exaumples shall more plainelye appere.

1. If you will diuide ¾ by ⅖, you muste sette the Diuisor (which is 2/3) next to the lefte hande, and the

diuidende ¾ toward your right hand, with a crosse betwene them: as may appere by this example in the margent 〈 math 〉〈 math 〉. Then you shal multiply y^e nu∣merator of y^e ⅔, which is 2. by y^e denominator of the ¾ which is 4. and and thereof commeth 8. which shalbe your new diuisor: set that 8. vnder the crosse, as the denominator, then multiply the numerator of the diui∣dende, that is to saye of the ¾ which is 3. by the denominator of the diuisor, that is to wit of the ⅔ which is 3. set y^t ouer the crosse, and it is 9. for the nu∣merator, which shalbe now the diui∣dende, or number to be diuided. Thē finally you shall diuide 9. by 8. and thereof commeth into y^e quotiēt 1. ⅛, and so oftentimes is ⅔ conteined in ¾. as doth appere before in the mar∣gent. But in case you woulde diuide ⅔ by ¾, you muste lykewise set your

diuisor ¾ next to your left hande, as is before saide. And then proceede, as is aboue declared, & you shal finde that ⅔ diuided by ¾ bringeth into the quo∣tient 8/9, which cannot be diuided nor abbreuied, wherfore it appereth that ⅔ diuided by ¾ bringeth but 8/9 of one vnitie into the quotient as doth ap∣pere.

〈 math 〉〈 math 〉

2. Likewise if you wil diuide a bro∣ken number by a whole number or else 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 〈 math 〉〈 math 〉 your diuisor, set y^t toward your left hande, and then multiply 13. by 4. accordīg to y^e first diuision, & there∣of commeth 52. for the de∣nominator,

set that vnder y^e crosse & multiply 3. by 1. which is 3. for the nu∣merator, that set ouer the crosse, and it is 3/52 as appeareth in the margent. But if you will diuide 13. by ¾ thē set the ¾ next your left hand and put one vnder 13. as in the last example, & it is 13/1 set y^t toward your right hande thus, as appeareth in the margent, 〈 math 〉〈 math 〉 and then worke according to y^e doc∣trine of the first diuision, & you shall finde that 13. being diuided by ¾ bringeth into y^e quo∣tient 52/3, then diuide 52. by 3. and therof commeth 17. ⅓, and so oftentimes is ¾ conteined in 13. as doth ap∣pere.

21
52
33	(17.••/••

3. And if you wil diuide whole nū∣ber by whole number and broken, or else whole nūber and brokē 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 35/6 for your

diuisor, then put 1. vnder 20. And it wyll bee 20/1, then shal you multiply 35. by 1. and 20. by 6. as is taught in the other diuisi∣ons, and you shall finde 120/35:〈 math 〉〈 math 〉 then di∣uide 120. by 35. and you shall fynde in your quotient 3. 3/7 & so many tymes is 5. ⅚ conteined in 20. as in the mar∣gent doth appere.

1

35

120 (3.••/7

35

But if you wyll diuide 5. 9/6 by 20. you must diuide 35. by 120, which you can not, wherefore you shall abbre∣uiate 35/126, and thereof commeth. 7/24.

4. If you will diuide a broken nū∣ber, by whole number and broken, or else 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

And they be 41/3 for your diuisor, then multiplye 41. by 4. & they make 164. for your denominator, likewise mul∣tiplye 3. by 3. and they make 9. for the numerator, and then will your sūme be 9/164. But if 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 num∣ber and broken, by whole number & broken, as to diuide 7. ¾ by 13. ⅔ you must reduce the whole numbers in∣to 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 set downe 41/3 to∣warde the left hande by∣cause it is your diuisor, and the 31/4 towarde the right hande, and multi∣plie 41. by 4. for your denominator, and thereof commeth 164. Likewyse multiply 3. by 3. for your Numerator, and it amounteth to 93. the whych diuision wyll bee thus 93/164 〈 math 〉〈 math 〉 as before

doth appere.

But if you will diuide 13. ⅔ 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 bro∣ken, must be diuided in such maner as broken numbers are, & there is no difference, sauing only that of many brokē numbers you must make but two broken numbers, that is to say 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 49/40 by the thirde reduction: and y^e 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 vn∣der the crosse and multiply 9. by 21: & thereof cōmeth 189. which set ouer y^e crosse for the numerator, 〈 math 〉〈 math 〉 and they

make 189/320 for this diuision as doth ap∣pere.

But if you woulde diuide 8/21 by 9/40. you must worke contrary to the laste example, that is to saye, you must di∣uide 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 shal diuide y^e same by ½: likewise if you will triple any fraction you must diuide it by 1/3. And for to quaduple any broken nū∣ber, you shall diuide it by ¼, and so is to be vnderstande of all other.

Example of Duplation.

IF you wil double ⅜ you shal diuide ⅜ by ½, and thereof com∣meth 6/8 〈 math 〉〈 math 〉, which being ab∣breuied are ¾: as by ex∣ample.

Or otherwise, in case the denomina∣tor of any fraction bee an euen num∣ber, you may take halfe the sayde de∣nominator, without anye other ope∣ration, 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 ⅜, 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 number, that is to say, not euen, thē you may multiply the numerator by 2. or else double y^e numerator, which is al one thing, and that fraction shal bee doubled. Example, if you will double ⅗ you must only multiply the numerator 3. by 2. & they be 6. which maketh that fraction to be 6/5, y^e which 6. being diuided by 5. bringeth 1. ⅕ and so much is the double of ⅗.

Example of Triplation.

If you will triple ⅗ you must diuide ⅗ by ⅕ and thereof commeth 9/5 whiche beinge diuided bryngeth 1 ⅘, or other∣wise, bicause the denominator is an odde number you may multiplie the numerator 3. by 3, and thereof com∣meth 9. which maketh 9/3 as before.

Example of quadruplacion.

If you will quadruple ⅘, you shall diuide ⅘ by ¼ and thereof commeth 16/5 which 16. being diuided by 5. bringeth 3 ⅕, or otherwise, bicause the denomi∣nator of y^e fraction is an odde nūber, you shall multiplie the numerator of the ⅘ that is to say 4. by 4. and therof commeth 16. the which diuide by 5. and you shall finde 3. ⅕ as before, and this sufficeth for duplacion, triplaciō and quadruplacion.

¶ The 9. Chapter treateth of the prooues of broken numbers. And first of Reductiō.

IF you doe abbreuiate y^e broken nū∣bers which bee reduced, you shall retourne them into their first estate: as by example, if you reduce ⅖ with ⅘ you shall fynde 10/15 and 12/15, then ab∣breuiate 10/15 and you shall finde ⅔, ab∣breuiate likewyse 12/15 and therof com∣meth ⅘ as before.

The proofe of Abbreuiation.

IF you doe multiplye that number which you haue abbreuied by that or those numbers, by the which you haue abbreuied them, you shall re∣turne them againe into theire fyrst estate. Example, if you will abbre∣uiate 32/48 by 16. in taking y^e 1/16 part both of the numerator, and also of the de∣nominator, you shall finde ⅔, y^e proofe is thus, you must multiplye both the numerator & denominator of ⅔ by 16. that is to say, three by 16. maketh 48. for y^e denominator, & 2. by 16. maketh 32. for the numerator, thē set the nu∣merator 32. ouer the denominator 48

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 y^e other, or others. Example: if you do adde ⅓ with ¼ you shall fynde 7/12. The proofe is, if you substract ⅓ from 7/12 you shall fynde re∣maining the other number which is ¼, or else if you doe substract ¼ frō 7/12 there wil remaine the other number, which is ⅓.

The proofe of Substraction.

If you do adde that number which remaineth, with the number whych you did substract, you shall fynde the totall summe, oute of the which you made y^e abatemēt: or otherwise, if you adde y^e two lesser numbers togither, you shal finde the greater. Example:

if you doe abate or substracte ¼ from ⅓ there wyll remaine 1/12. The proofe is thus: you must adde 1/12 & ¼ togither, and you shall fynde ⅓, which is the greatest number.

The proofe of Multiplication.

If you diuide the producte of the whole multiplication, by the multi∣plicator, you shall fynde in your quo∣tient, the multiplicande or number by the which you haue multiplyed: or else if you diuide the total sum which is come of the multiplication, by the multiplicande: you shall finde in the quotiēt the multiplicator. Example, if you multiply ⅔ by ⅘, the product of this multiplication will bee 8/15. The proofe is thus: you shal diuide 8/15 by y^e multiplicator ⅘, and therof cōmeth ⅔. Or els diuide 8/15 by ⅔ & you shal finde the ⅘ which is the multiplicator.

The proofe of Diuision.

If you doe multiplye the quotient by the diuisor, you shall finde the nū∣ber which you did diuide, y^t is to say, your diuidende. Example: if you di∣uide ⅔ by ¾, your quotient wil bee 8/9 y^e proofe is thus, you must multiply 8/9 by ¾, and therof commeth 24/36 which be∣ing abbreuiated are ⅔ whiche is your diuidende, & by this maner all whole numbers haue theire proofes as well as broken numbers.

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

[unspec I] FInde two numbers, wherof the 2/7 of the one number may bee egal vnto the ⅜ of y^e other. Aunswere: you shal reduce 2/7 & ⅜ crosse∣wise, and you shall fynde 16. ouer the 2/7 and 21. ouer the ⅜, which are y^e two numbers that you seeke: for y^e ⅜ of 16. are 6. and so are the 2/7 of 21. lykewise 6. wherefore you may perceaue that

the ⅜ of 16. which are 6. are egall vnto the 2/7 of 21. which is also 6.

2. Finde two numbers, wherof y^e ⅔ 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 twoo numbers y^t you seeke. For y^e ⅔ of 3. which is 2. is double vnto the ¼ of 4. which is but 1.

3. Finde two numbers wherof the ⅓ 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 ⅕ togither, & they are 9/20, thē reduce 7/17 & 9/20 crosse∣wise, & you shall haue 140. ouer the 7/12 & 108. ouer the 9/20, which are the two nūbers that you seeke. For 63. which are the 7/12 of 108. are also the 9/20 of 140.

4. Finde two numbers, wherof y^e ½ the ⅓ and the ¼ of the one of them, may be egall vnto the ⅕ the ⅙ and 1/7 of the other number. Aunswere: first you

must adde ½, ⅓ and ¼ togither, & they make 13/12: then adde ⅕, ⅙ and 1/7 togi∣ther, & they make 107/210. Then reduce 13/12 and 107/210 crossewise, as by the fyrst question of reduction, and you shall finde 2730. ouer the 13/12 and 1284. ouer the 107/210, which are the two numbers that you seeke: for 1391 which is the ½ the ⅓ and ¼ of 1284. is lyke to the 11/56 & 1/7 of 2730, which is also 1391.

5. Finde three numbers, wherof the ⅖ of y^e first, the 3/7 of the seconde, & the 4/9 of the thirde, may be egall the one to the other. Aunswere: set downe the 23/57 and 4/9, and then multiplie the Denominator of the ⅖ that is to say 5. by the Numerators of the other two Fractions, that is to say, by the Nu∣merator of 3/7, and by the Numerator of 4/9, which is 3. and 4. And thereof commeth 60. for your fyrst number, then shall you multiplye the Deno∣minator of the 3/7 which is 7. by the Numerators of ⅕ and 4/9, that to to say by 2. and 4. and thereof commeth 56.

for the seconde number. Then mul∣tiplie the Denominator of 4/9, that is 9. by the Numerators of ⅖ and 3/7 that is by 2. and by 3. and therof commeth 54. for the thirde number.

And thus the ⅖ of 60. which is 24. is likewise the 3/7 of 56. which is the se∣cond number: and the 4/9 of 54. which is the thyrde number.

6. Finde three numbers, of which the fyrste and the seconde may bee in such proporcion as ½ and ⅓, and the se∣conde 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 ⅕. Then say by the Rule of three, if 5. do gyue mee 4. what shall two gyue mee, which is the seconde proportio∣nal, multiply the seconde number 4. by the thyrde number two, and ther∣of commeth eyght, the which diuide

by the first number 5. and therof com∣meth 1. ⅗ for the thirde proportionall, and you shal fynde that 3.2.1. ⅗ are the three numbers proportionall that I demaunde, or else 15.10. & 8. in whole numbers.

1. What number is that, vnto the which 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. Aunswere:: abate ⅖ from ⅚, and there will remaine 13/30, which is the number that you desyre.

3. What nūber is that, whereun∣to if you adde 7. ⅔, the whole additiō will be 12. ¼. Aunswere: abate 7. ⅔ frō 12. ¼. & the remaine wil be 4.7/12 which is the number y^t you desire to know.

4. What number is that, where∣unto if you adde the ¾ of it selfe, that is to say, of the nūber that you seeke, the whole addition may be ⅚.

Aunswere: Here foloweth a generall rule for all such like questions. Fyrst, of 3. which is y^e numerator of ¾ make still the numerator and likewise of 3. and 4. togither, which is both y^e nu∣merator, and the denominator of the ¾ make your denominator, so you shal finde 3/7, then take the 3/7 of ⅚ which is 15/42 or 5/14, and substract them from ⅚, and there wyll remaine 10/21 which is the number that you seeke.

5. What number is that, vnto the which if you adde his owne ⅔ y^t is to say ⅔ of it selfe, y^e whole addition shall bee 20. Aunswere: doe as in the laste question: of the numerator of ⅔, that is to saye, of 2. make still your nume∣rator. And likewise of y^e numerator. 2. and the denominator 3. of the ⅔, make of them both, your denominator, and you shall finde ⅖, thē take the ⅖ of 20. which are 8. And abate them from 20. and there will remaine 12. which is the number that you desyre and so is to bee done of all suche lyke rea∣sons.

1. What number is that, from the which if you do abate 17. the rest may be 19. Aunswere: adde 17. and 19. togi∣ther and you shall finde 36. which is the number that you seeke.

2. What number is that, from the which if you abate ⅗ the rest maye be ⅛. Aunswere: adde ⅗ and ⅛ togither: and you shall finde 29/40 whiche is the number that you demaunde.

3. What number is that, from the which if you deduct 13. ½ the rest maye be 5.5/7. Aunswere: adde 13. ½ and 5.5/7 to∣gether, and thereof commeth 19.3/14, which is the number that you seeke.

4. What number is that, from y^e which if you substracte hys ⅖ the rest may be. 12. Aunswere: & a rule for such like reasons, y^t is to saye frō the deno∣minator of ⅖ which is 5. abate 2. which is his numerator, and there resteth 3. for y^e denominator, and thus of ⅖ you haue now made ⅔, then take the ⅔ of 12. which are 8: and adde them vnto

12. and thereof commeth 20. for the number which you desyre.

5. What number is that, from the which if you doe abate hys ¾, the rest maye be 8/9. Aunswere: from the De∣nominator of ¾ which is 4. substracte hys Numerator 3. and there resteth 1. Thus of ¾ you haue made 3/1. Then multiplye 3/1 by 8/9, and thereof com∣meth 2 ⅔, the which adde vnto 8/9, and you shal haue 3.5/9, which is the num∣ber that you seeke.

6. What number is that, from the which if you abate his ⅘, the rest maye be 12. ⅔. Aunswere: Doe as you dyd in the laste question, and you shal fynde that the ⅘ wil bee 4/1. And ther∣fore multiplie 12. ⅔ by 4/1, and thereof commeth 50. ⅔, the which adde vnto 12. ⅔, and you shal fynde 93. ⅓, for the number that you demaunde. And thus of all such lyke questions.

1. What number is that, which being multiplied by 13. the whole.

Multiplication shall mount to 221. Aunswere: diuide 221. by 13. and there∣of commeth 17. which is the number that you seeke.

2. What number is that which being multipled by 15. y^e whole mul∣tiplication wil amoūt to ¾. Aunswere: diuide ¼ by 15/1 and therof commeth 1/20 which is the number, that you seeke.

3. What number is that, which being multiplyed by 21. the whole multiplication wil be 16 ⅘. Aunswere: diuide 16. ⅘ by 21/1, and you shall finde ⅘ whych is the number that you de∣maunde.

4. What number is that, which being multiplyed by ¾ the multipli∣cation will amount to 18. Aunswere: diuide 18/1 by ¾, and thereof commeth 24. which is the number that you de∣syre to knowe.

5. What number is that, which if it be multiplied by ⅔ the whole mul¦tiplication wyll bee ¼. Aunswere: di∣uide ¼ by ⅔ and the quotient will bee ⅜

which is y^e number that you require to know.

6. What number is that, whych beyng multiplied by ⅝, the product of y^e multiplicatiō wil be 16. ⅔. Aunswere: diuide 16.2/3 by ⅝ and thereof commeth 26. ⅔ which is the nūber y^t you seeke.

¶Here ensueth other necessarie Questions, which are wrought by Multiplicatiō in broken nū∣bers.

[unspec I] I Demaunde how much the ⅝ of 20. shillings are worth, or what are y^e ⅝ of 20. shillings. Answere: you must multiplye ⅝ by 20/1 and y^e pro∣duct wil be 100/8, therefore diuide 100. by 8. and therof commeth 12. ½, which is to saye, 12.s. 6.d. and so much are y^e ⅝ of 20. shillings worth.

2. I demaunde what the ¾ of ⅚ of a pounde of money are worth, that is to say of 20. s. Auns. multiplye ¾ by ⅚. And thereof commeth ⅝. Then take

the ⅝ of 20. shil. as in the last question going before, and you shall finde 12.s 6. pence, and so much are the ¾ of ⅚ of 20. s. worthe.

3. I demaunde what the ⅔ of 8. d. ½ are worth. Aunswere: multiply 8.½ by ⅔ or else ⅔ by 8. ½, which is all one, and you shall fynde 34/6. Then diuide 34. by 6. and your quotient wyll be fyue pence ⅔, and so muche are the ⅔ of 8. pence ½ worth.

4. What are the ¾ of 14. pence ⅗. Aunswere: multiplie 14 ⅗ by ¾, and thereof commeth 219/20. Therefore di∣uide 219. by 20. and your quotient wil be ten pence 19/20, and so much are the ¾ of 14. ⅗.

5. How many quarters or fourthe partes are contayned in 7.⅔. Auns. multiply 7.⅔ by 4/1 (bicause one whole contayneth 4. quarters) and thereof commeth 30.⅔, and so many quarters are in the 7.⅔, that is to say 30. quar∣ters, and ⅔ of a quarter.

6. Howe many thirdes are in ¾

and ½, that is to saye in 3. quarters, and ½ of one quarter, which are ⅞ by the fifte reduction. Aunswere: multi∣plie ⅞ by 3/1 (for bicause that in 1 whole are conteined 3. thirdes) and thereof will come ⅔ and ⅝ of a thirde, and so many thirdes are in ¾ and ½ or in ⅞, which is all one.

¶Questions done by diuision in broken number.

1. What number is that, which being diuided by 17. the quotient wil be 13. Aunswere: multiplye 17. by 13. And thereof commeth 221. whiche is the number that you seeke.

2. What number is that, which being diuided by ¾ the quotiēt wil be 21. Aunswere, multiply 21/1 by ¾ & there∣of commeth 63/4. Then diuide 63. by 4. and thereof commeth 15. ¾, which is the number that you seeke.

3. What number is that, whych being diuided by ⅛, the quotient

wilbe ⅔. Aunswere: multiplye ⅔ by ⅓ & thereof commeth 2/24 which being ab∣breuiated are 1/12 for y^e number which you require.

4. What number is that, whiche being diuided by ⅘ the quotient will be 16. ⅔? Aunswere: multiply 16. ⅔, by ⅘, and thereof commeth 200/15. There∣fore diuide 200. by 15. and thereof com∣meth 13. ⅓ which is the number that you desire to finde.

5. What number is that, which be∣ing diuided by 13 ⅓, the quotient will be 20. Aunswere: multiplie 20/1 by 13 ⅓, and therof commeth 800/3, then diuide 800. by 3. and thereof commeth 266. ⅔, for the number, which you seke.

6. What number is that, which if it be diuided, by 12. ½ the quotient will be ⅞ Aunswere: multiplie by 12 ½ and therof commeth 175/16, then diuide 175 by 16. and thereof commeth 10.15/16, for the nūber which you desyre.

¶Other necessarye questions done by Diuision in broken number.

[unspec I] I Demaunde what part 30. is of 70. Aunswere: diuide 30. by 70. which you can not, for they are 30/70, but abbreuiate them & they are 3/7. Thus 30. are the 3/7 of 70.

2. I demaunde what parte 10. is of 16 ⅔. Aunswere: diuide 10/1 by 16. ⅔, and thereof commeth 30/50 which being ab∣breuiated are ⅗. And thus 10. is found to be ⅗ of 16 ⅔.

3. More, what part is 25. of ⅝. Auns. diuide ⅝ by 25/1, and thereof commeth 5/200, which being abbreuiated is, 1/40. And thus ⅝ is but the 1/40 of 25.

4. More, ⅚ what part are they of ⅞. Aunswere: diuide ⅚ by ⅞, and you shal finde 40/42 which abbreuiated are 20/21.

5. More, ⅘ what parte are they of 13. 1/3. Aunswere: diuide ⅘ by 13. ⅓ and you shall fynde 12/200, which being all bre∣uiated

are 3/80. And thus 4/1 are the 5/50 of 13. ⅓.

6. More 12. ½ what part are they of 30. Aunswere, diuide 12. ½ by 30/1, & you shal find 25/60 which being abbreuiated are 5/12 and thus 12. ½, are the 5/12 of 30.

7. More, 16 ⅔ what parte are they of of 57 1/7. Aunswere, diuide 16. ⅔ by 57 1/7 and thereof commeth 350/1200 which be∣ing abbreuiated are 7/24, and thus 16. ⅔ are the 7/24 of 57.1/7.

8. More, ¾ and ⅔ of ¼, or 3. quarters & ⅔ of one quarter, what parte are they of 1. Aunswere, reduce 5/4 and the ⅔ of ¼ into one broken by the first reductiō, and you shall finde 11/12. And thus the ¾ and ⅔ of ¾ are the 11/12 of one whole.

9. More, of what number are 9. the 2/3. Aunswere, diuide 9. by ⅔, & thereof commeth 13. ½, which is the number wherof 9. are the ⅔.

10. More of what number are ⅖ the ¾. Aunswere, diuide ⅖ by ¾, and thereof commeth 8/15 which is the number wherof ⅔ are the ¾ of the same nūber.

11. More, of what number are 5. ¾ the 3/7. Aunswere: diuide 5. ¾ by 3/7, and you shall fynde 13.5/12 which is the nū∣ber whereof 5. ¾ are the 3/7.

12. More, 9 ⅔ what part are they of 33. ½. Aunswere: diuide 9 ⅔ by 33. ½. And thereof com∣meth 58/201: and thus 9. ⅔ are the 58/201 of 33. ½ as appea∣reth.

The thyrde parte treateth of certaine briefe rules, called rules of practise, with di∣uers necessarie questions pro∣fitable for Marchaunts.

The first Chapter.

SOme ther be, which doe call these rules of practise briefe rules, for that by them ma∣ny questions may be done wyth quicker expedition, than by the rule of three. There be others which call them the small multiplication, for bicause that the product, is alwaies lesse in quan∣titie, than the number whiche is to bee multiplied. This practise com∣meth not in vse, but onely amonge small kindes of numbers, whiche haue ouer them, other numbers that are greater. And thys beyng well considered, is no other thyng

but to conuert lesser and particu∣ler kindes of number, into greater, the which maye be done by the mea∣nes of diuision, in taking the halfe, the thirde, the fourth, the fift, or suche other partes of the summe, which is to be multiplied, as the multiplier is part of hys greater kinde, and that which commeth thereof is worth as muche (not in quantitie, but in his owne forme) as if you did multiplye simplie the two summes, the one by the other: And for the better vnder∣standing of suche conuersions, you must haue respect to one of these two considerations. The first is, whē one woulde demaunde this question. At 6.d. the yarde of Cotton, what are 18. yardes worth by the price? It is ma∣nifest that they are worth 18. peeces of 6. pence the peece, or 18. half shillings, which must be turned into shillings, in taking y^e halfe of 18.s. & they make 9.s. Or otherwise you must cōsider, that at 1.s. the yarde, the 18. yardes

are worth 18.s. wherefore at 6.d. they shall be but halfe so much, for 6.d. is but the ½ of 1.s. Therefore you must take the ½ of 18. and they make 9.s. which are worthe as much as 108.d. that is to say, as 18. times 6. pence.

2. First, if you will multiplie anye number after thys maner by pence whereof the number of y^e same pēce, doe not extende vnto 12. and therof to bring shillings into the product: you must know the certaine partes of 12. which are these: that is to say, 6.4.3. 2. and 1. For 6. is the ½ of 12. and 4. is the ⅓ of 12:3. is y^e ¼:2. is the ⅙: and 1, is the 1/12. Then for 6.d. which is the halfe of 1. shilling, you must take the ½ of al the number which is to be mul∣tiplyed. And that which commeth thereof, shal be shillings, if there doe remaine 1. it is 6. pence.

For foure pence you muste take the ½ of all the number that is to bee multiplyed: and if anye vnities doe remaine, they shall bee thyrdes

of a shilling, euery one being in value 4. pence.

For 3. pence you must take the ¼ of al the summe: if anye vnities doe re∣maine, they shall bee fourthes of a shilling, euery one being worth thre pence.

For 2. pence you must take the ⅙ of all the summe, and if any vnities doe re∣maine, they shall be sixte partes of a shilling, being euerye one of them worth two pence.

For d. take the 1/12 of the whole sūme, if anye vnities remaine, they are 12. partes of a shilling, eche of them be∣ing in value 1.d. as by these examples following doth plainely appere.

j.

At 6. Pence the yarde.

What 59. yardes.

29. shill. 6. Pence.

ij.

At 4. Pence.

What 82.

27. shill. 4. Pence.

iij.

At 3. Pence.

What 927.

24. shill. 3. Pence.

iiij.

At 2. Pence.

What 346.

57. shil. 8. Pence.

v.

At 1. Pence.

What 343.

28. shil. 7. Pence.

Here you may see in the fyrst exam∣ple that 59. yardes, at 6. pence y^e yarde is worth .29. shil. 6.d. in taking the ½ of 59. And in the seconde example, y^e 82. yardes at 4. pēce the yarde, is worth, 27.s. 4.d. in taking the ⅓ of 82.

Likewise, in the thyrde example, 97. yardes, at three pence the yarde, bringeth 24. shil. 3. pence, in taking the ¼ of 97. Also in the fourth exam∣ple, 346. yardes, at 2. pence the yarde, maketh 57. shillings eyght pence in taking the ⅙ of 346. And fynal∣ly

in y^e fift example .343. yardes, at 1.d. the yarde, amount to 28. shil. 7.d. in taking the 1/12 of 343. And so is to bee done of all such lyke, when the num∣ber of the pence, is any of the certaine partes of 12.

But if the number of the pence be not a certayne parte of 12. you muste reduce them into some certayne par∣tes of 12. and after the foresayd maner you shal make two or three productes as neede shal require, and adde them togyther into one summe as 5.d. may be reduced into 4. & 1. or els into 3. & 2: wherfore if you wil work by 4. & by 1: you must for 4.d. take fyrst the ⅓. of y^e number, that is to be multiplied, and for 1.d. take the 2/12, or rather for 1.d. ye may take the ¼ of the producte which did come of the 4.d. bycause that 1.d. is the ¼ of 4.d. But if you wyl worke by 3, and 2, you shal take for 3.d. the ¼. of the number which is to bee multi∣plied: and likewyse for 2.d. the ⅙ of the same number, adding togyther both

the productes. The totall summe of those two numbers shall be the solu∣tion to the question. And in like ma∣ner is to be done of all other. As by these formes folowing may appeare.

j.

At 5. d. the yarde.

What 49. yardes?

16. shil. 4. d.

4. shil. 1.

20. shil. 5. d.

ij.

At 7. d.

What 5••?

18. shil. 0

13. shil. 6

31. shil. 6. d.

iij.

At 8. d.

What 40?

13. shil. 4

13. shil. 4

26. shil. 8. d.

iiij.

At 9. d.

What 73?

36. shil. 6

18. shil. 3

54. shil. 9. d.

v.

At 10. d.

What 32?

16. shil. 0

10. shil. 8

26. shil. 8. d.

vj.

At 11. d.

What 27?

9. shil. 0

9. shil. 0

6. shil. 9

24. shil. 9. d.

Here in this same first exaumple where it is demaunded (at 5. pence y^e yard) how much are nine and fourty yardes worth? Fyrst for foure pence,

I take y^e ⅓ 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. shil. 1.d. these twoo sūmes added togither, do make 20.s. 5.d. And so much are the 49. yar∣des worth at 5.d. the yard.

For 7.d. take the ⅓ and the ¼ of the whole summe which is to be multi∣plied, and adde them togither, that is to say, for 4.d. the ⅓ and for 3.d. the ¼: bycause 4.d. is the ⅓ of 12.d. and 3.d. is the ¼ as in the second example before doth appeare: Where the questiō is thus, at 7.d. y^e yard what are 54. yar∣des worth? Fyrst for 4.d. I take the ⅓ of 54: and they make 18.s. Likewyse for 3.d. I take the ¼ of 54. and they are 13.s. 6.d. Then I adde 18.s. and 13.s. 6. d. togither, so both amount to 31.s. 6.d and so much are the 54. yardes worth at 5.d. the yarde.

Otherwyse for 7.d. take first the ½, of the whole sūme for 6.d. Then for 1.d. take the ⅙ of thesame product, and

adde them togither, so shall you haue the lyke summe as before.

For eight pence you must first take ⅓ of the whole sūme for 4. pence, and another ⅓ for other 4.d. and adde thē togyther as in the example doth eui∣dently appeare. Where the question is thus, at 8.d. the yarde, what are 40 yardes worth? Fyrste 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 whiche is also 13. shillings & 4. pence. These two summes being added to∣gither, do make 26. shillings 8. pence, and so much are the 40. yards worth at 8. pence the yard as in the third ex∣ample abouesayd doth appeare.

Otherwayes, for eyght pence you mai take first the ½ of the whole sūme for 6.d. Thē for 2.d. you shal take the ⅓ of the product, which did come of the sayd ½, and adde them togither, so shal you haue likewise the solution to the question. As in the same third exāple of 40. yardes, I take first the ½ of 40.

for 6.d. and thereof commeth 20. shil. then for 2.d. I take ⅓ of the saide pro∣duct, that is to say of 20.s. which brin∣geth 6.s. 8.d. these two summes (20.s and 6.s. 8.d.) I adde togither & they make 26.s. 8.d. as before.

For 9.d. you must take the ½ & the ¼ of the whole sūme, and adde them togither: or else for 6.d. take fyrst ½ of the whole summe, then for 3.d. take y^e ½ 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 exam∣ple, 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 two summes doe I adde togither, & they make 54. shil. 9.d. as in y^e saide fourth example is euident.

For 10.d. take first the ½, then y^e ⅓ of the whole sūme, & adde thē together

For 11.d. take fyrst ½ for 4. pence, se∣condely,

another ⅓ for other 4.d. and thirdely ¼ for 3.d. of all y^e whole sūme: and adde them togither.

Or else for 11.d. take first the ½ then the ⅓ of the whole summe, and final∣lye the ¼ of the laste product, adding them togither.

3. Lykewise by the same reason, when you wil multiply (by shillings) anye number 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 ⅕:2. 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 ½ of the number, which is to bee multiplied, and you shall haue poundes in y^e pro∣duct. If there doe remaine 1, it shalbe worth ten shillings.

For 5. shillinges you muste take the ¼ of the number whiche is to bee multiplied, & if there do remaine any vnities, they shall be foure partes of

a poūd, euery one being in value 5.s.

For 4.s. you must take the ⅕ of the number which is to bee multiplied. And if there do remaine any vnities, they shal be fift parts of a pound eue∣ry one being worth four shillings.

At 10. shillings the Peece.

What 75. Peeces?

37. li. 10. shil.

At 5. shil.

What 89.

22. li. 5. shil.

At 4. shil.

What 93.

18. li. 12. shil.

For 2. shillings you must take the 1/10 of the nūber that is to be multiplied. Wherefore, if you wyll take the 1/10 of any number: you muste seperate the last figure of the same number which is nerest your ryghte hande, from all

the other fygures. For all the other figures which doe remayne towarde your lefte hande, from the same fy∣gure, which is seperated, shall bee the sayde 1/10 of a pounde: and that sepera∣ted fygure, towarde your right hand shall be so many peeces of 2. shillings the peece: the which fygure muste be doubled, to make therof shillings, as by these examples appeareth.

At 2. shil.

What 9/8.

9. li 16. shil.

At 2. shil.

What 40/3.

40. li. 6. shil.

Herevpon dependeth another exact way for to multiply by shillings (if y^e number of shillings be euē) which is thus: you shal take ½ the nūber of the same shillings, and conuert them in∣to peeces of 2. shillings. Then by the

number of this halfe, you must firste multiply the last figure toward your right hande, of the nūber which is to be multiplied: And if ther be any ten∣nes in the same product, those must ye reserue in your minde: But if (wyth the same or els without the same) you doe finde any diget number, the same diget number shall you double, & put it in the place of shillings: Thē muste you proceede to the multiplication of the other figures, adding vnto y^e pro∣duct the tennes which you before re∣serued: and therof shal come pounds.

Now, for your better vnderstan∣ding of this which hath bene said and by the way of example, I wil propone vnto you this question.

At 8. shillings the grosse, what are 97. grosse worth after the rate?

Firste in this example I take halfe the number of Shillinges, as before is taught, that is to say of eighte shil∣lings, which is foure shillinges: this 4. shil. I put apart, behinde a crooked

line righte againste 97. towardes the left hand, as here you may see and as here after apereth by diuers exāples.

At 8. shil. the Grosse.

4) What 9/1

38. li. 16. shil.

At 6. shil.

3) What 9/9

29. li. 14. shil.

At 12. shil.

9) What 34/5

207. li. 0 shil.

At 14. shil.

7) What 21/0

127. li. 0. shil.

Now in the first example, where it is demaūded, at 8.s. the grosse, what are 97 grosse? First the ½ of 8.s. which is 4.s. being set apart behind the cro∣ked line, as before is sayd: thē I mul∣tiplye y^e 97 by 4. saying first, 4. times 7. is 28. I double y^e diget nūber 8. and

that maketh 16, the which 16, I do put vnder the line, in the place of shillīgs & I kepe y^e 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: wher∣vnto I adde the 2, tennes, which be∣fore I dyd reserue, and they make 38. Therefore I put 38, vnder the lyne in the place of poundes, and the whole summe wil be 38. li. 16.s. Thus much are the 97. grosse worth, at eight shil∣lings the grosse: the like is to be done of all other. As of 12. shillings in mul∣tiplying by 6. Likewise of 6. shillings if you multiply by 3, also of 14. if you multiply by 7. And so of all euen nū∣bers after the same manner.

For 1. Shilling you must take the ½ of the 1/10 parte of any number that is to be multiplied.

At 1. shil.

What 35/0

17. li. 10. shil.

And if any thyng do remayne, they are shil. Thus by this manner shil.

are conuerted into poundes: for it is euen lyke, as if you did diuide thē by 20.s. as by this exāple in the margent doth appeare. Wher it is demaūded at 1.s. the yard, the peece, or any other thing, what are 350. worth?

First I seperate the laste fygure of 350. nexte to my ryght hand, which is the 0 wyth a line betweene it and the figure 5. Then I make a line vnder the 35/0, and I take the ½ of 35, after this maner: saying the ½ of 3. is 1. and 1. remayneth, which remayn signifieth 10. in that second place. Then I put 1. vnder the line agaynst 3, & I proceede to the rest, saying: the halfe of 15, is 7. (which 15. came of the 1. that remay∣ned, and of the 5. in y^e first place) I put 7. vnder the line right agaynst 5, and they make 17. li. The 1, which did last remayne, is 10.s. Therfore I put 10.s aparte vnder the line, and the whole summe is 17. li. 10.s. so muche are 350. worth at 1.s. the peece.

But when the number of shillings

is not some certayne parte of 20. shil. you muste then conuert the same nū∣ber of shillings, into the certayn par∣tes of 20. and make twoo or three pro∣ducts, as nede shal require, the which muste bee added togyther after thys maner following.

For 3. shillings you must firste take for 2. shil. the 1/10 of the number that is to be multiplied, then for 1. shillīg you must take the ½ of the producte which did come of the same 1/10 part: and adde those twoo sūmes togither, as appea∣reth by this example following.

At 3.s. the peece of any thing, what shall 684 peeces coste mee after the rate. Firste, for 2 shillings I take the 1/10 of 684, which is 68:

At 3. shil.

What 68/4?

68. li. 0. sh

34. 4.

102. li. 12. sh

in seperating the laste figur 4, which I must double, and they be 8. I set eyghte 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,

for the 2.s. secondlye, for 1. shil. I take the ½ of the product, that is to saye: of 68. li. 8.s. which is 34. li. 4.s. and I put the same vnder the 68. li. 8. shil. Then finally, I adde those two sum∣mes together, y^t is to saye, 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 y^e peece, as may appere in the margent.

For 6. shil. take 3/10 of the number which is to be multiplied: that is to say, first 1/10, then double the product of the same 1/10 and adde them togither. Or otherwise for 4.s. take fyrst y^e ⅕ of y^e nūber that is to be multiplied, thē take the ½ of the product which is for two. s. and adde them togither.

Or else take for 5. shil. the ¼ of the whole summe, then for 1. shil. the ⅕ of the product and adde them togither.

Likewyse for 7. shil. take fyrst for 5. shil. the ¼ then for ••. shillings take the 1/10 of the number which is to be mul∣tiplied, and adde them togither.

For eyght shillings take the ⅖ at two sundry tymes, that is to say, first ⅕ for 4. shil. and then as much more for o∣ther 4. shil. and adde them togyther.

For 9. shil. take first the ¼ and lyke∣wise the ⅕ of the number 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 ⅕ part of the pro∣duct, and adde them togither.

For 13. shil. take the ¼ then the ⅕, & agayne another ⅕ of the nūber which is to be multiplied. And adde the pro∣ductes togither, that is to say: fyrste for 5. shil. take the ¼, then for 4. shil. take the ⅕. And agayne, another ⅕ for the other 4. shil. and assēble the three productes, the like is to be done in al others, when the price of the thing which is valued, is onely of shillings. And as by these examples following doth playnly appeare.

At 6. shil.
Whas 67.
13. .8
6. .14
20. li. 2. shil.
At 7. shil.
What 347.
86. .15
34. .14
121. li. 9. shil.
At 8. shil.
What 540.
108. .0
108. .0
216. li. 0. shil.
At 9. shil.
What 230.
57. .10
46. .00
103. li. 10. shil.
At 11. shil.
What 159.
79. .10
7. .19
87. li 9. shil.

At 12. shil.
What 349.
174. .10
34. .18
209. li 8. shil.
At 13. shil.
What 267.
66. .15
53. .8
53. .8
173. li. 11. shil.

4. Likewise in multiplying by pence you shal haue (at the first instāt) poū∣des in the product, in case you knowe the certayne partes of the 1/10 of a poūd or of 24. pence, which are these 12, 8, 6, 4, 3, and 2. For 12, is the ½ of 24: 8. is y^e ⅓: 6 is the ¼: 4 is the ⅙: 3 is the ⅛: and 2. the 1/12: but for 12.d. which is 1. shil. we haue before made mencion thereof.

For 8.d. you muste take the ⅓ of the 1/10 and the rest which are the peeces of 8.d. must be doubled to make of them peeces of 4.d. And of the same num∣ber being doubled, you must take the

⅓ which wil be shillinges, & if there do yet remayn any thing, they are thirds of a shilling beeing in value 4. pence the peece.

For 6.d. take the ¼ of the 1/10, and of that which remayneth you must take the ½ which shall be shillings, if there do yet remayne 1, it shall bee in value 6. pence.

For 4.d. you must take the ⅙ of the 1/10 and of that which resteth, take the 1/•• to make therof shillings, if any thing do yet remayne, they are thirdes of a shilling, being in value 4.d. the pece.

For 3.d. take the ⅛ of the 1/10, and of that which remayneth, take the ¼, to make of them shillings: if any thing do yet remaine, they are fourthes of a shilling, euery one of them beeing worth 3.d.

For 2.d. take the 1/12 of the 1/10: and of that which resteth take y^e ⅙ the which are shillings, if there do still remayne any thing, they shall be sixt parts of a shilling, euery one being in value 2.d

For 1. d. it is not possible with ease, to bring of pence, poūds (into the pro∣duct) vpon the total summe: But first you must bring thē into shillings by y^e order of the secōd rule of this chapter, and then afterward you shal conuert thē into poūds, if nede so require. As by this exāple following may apeare.

At	8. d.
What	59/5.
	19. li. 17. shil. 4. d.
At	6. d.
What	67/8.
	16. li. 19. shil.
At	4. d.
What	93/4.
	15. li. 11. shil. 4.d.
At	3. d.
What	57/1.
	7. li. 2. shil. 9.d.
At	2. d
What	36/4.
	3. li. 0. shil. 8. d.

At	1. d.
What	66/5.
	5••5.	00. shil.	4.d.
	2. li.	16. shillings.	4.d.

But if the number of pence, be not a certayne parte of 24. pence. Then must you bring them into the certain partes of 24. and make therof diuers productes, which must be added togi∣ther, as shall hereafter appeare.

For 5. pence you shall firste take for 3. pence, then for 2. pence, and adde them togither, according to the in∣struction of the last rule. Or else firste take for 4. pence, and then for 1. d.

For 7.d. firste take for 4. d. then for 3. d. and adde them togither:

For 9. d. fyrste take for 6.d. then for 3. d. adding them togither.

For 10. d. fyrste take for 6. d. then for 4. d. and adde them togither.

For 11. d. take first for 8. d. then for 3. d. & adde them togither: as by these examples following doth appeare.

At	5. d
What	92/7.
	11.	.11.	.9
	7.	.14.	.6
	19. li.	6. shil.	3. d.
At	7. d.
What	51/2.
	8.	.10.	.8
	6.	.8.	.0
	14. li.	18. shil.	8. d.
At	9. d.
What	54/6.
	13.	.13.	.0
	6.	.16.	.6
	20. li.	9. shil.	6. d.
At	10. d.
What	27/3.
	6.	.16.	.6
	4.	.11.	.0
	11. li.	7. shil.	6. d.
At	11. d.
What	26/4.
	8.	.16.	.0
	3.	.6.	.0
	12. li.	2. shil.	0. d.

5. If you wil multiply any number by shil. and pence, being both togither you must take first for the shil. accor∣ding to y^e instructiō of the rule of this first chapter, then take for y^e pence af∣ter the order of the fourth rule before mencioned: but if ther be any certayn partes of 1. li. cōtayning both shil. and pence, thē for such parts you shal take the like part of the number that is to be multiplied, as the nūber is part of 1. li. the which certain parts are these, 6. s. 8. d: 3. s. 4. d: 2. s. 6. d: & 1. s. 8. d. For 6. s. 8. d. is the ⅓ of a li. 3. s. 4. d. is y^e ⅙ of a li. 2. s. 6. d. is the ⅛: & 1. s. 8. d. is the 1/12. then for 6. s. 8. d. you must take the ⅓ of the nūber that is to bee multiplied: & if ani thing do remain, they are thirds of a li. euery one being worth 6. s. 8. d

For 3. s. 4. d. you must take y^e ⅙ if ani thing do remain, they are sixt parts of a li. euery one being in value 3. s. 4. d.

For 2. s. 6. d. you must take the ⅛: if any thing be remaining they ar eight parts of a li. ech one beīg worth 2.s. 6.

For 1. shil. 8. d. you shal take the 1/12 if there do any thing remaine, they are twelfth partes of a pounde euery one being valued at 1. shil. 8.d.

At 6. shil. 8. d

What 647.

215. li. 13. shil. 4. d.

At 3. shil. 4. d

What 220.

36. li. 13. shil. 4. d

At 2. shil. 6. d

What 47.

5. li. 17. shil. 6. d

At 1. shil. 8. d

What 400.

33. li. 6. shil. 8. d

6. Heere shall you accustome youre selfe, to multiply by all sortes of sum∣mes, being compoūd of shillings, and pence, which mai come to practise. As thus, for 1. s. 1. d. for 1. s. 2. d. 1. s. 3. d. for 1. s. 4d. Likewise for 2.s. 1.d. 2.s. 2.d. 2.s. 3. d. 2. s. 4. d. And so of all other: consi∣dering moreouer, many subtile ab∣breuiations, which happen oftētimes

that are easy to be cōceyued. As thus at 11. s. 3. d. after that I haue takē first the ½ for 10. s. Then for 1. s. 3. d. I take the ⅛ of the product, bycause 1. s. 3. d. is the ⅛ of 10. s. in taking the sayd ⅛ of the product. And by this meanes, whē ye haue taken one product, ye may oftē∣times vpō y^e same, take another more briefly thā vpō y^e sūme y^t is to be mul∣tiplied, which thing you must foresee.

At 11. shil. 3. d.
What 53.
26. .10. .0
3. .6. .3
29. li. 16. shil. 3. d.
At 6. shil. 3. d.
What 58.
14. 10. .
3. 12. 6.
18. li. 2. 6. d.
At 12. 8. d.
What 64.
32. 0. .
6. 8. .
2. 2. 8.
40. li. 10. shil. 8. d.

7. But if you wil multiply, by poūds, shillings and pence being altogither. Firste you muste wholy multiply by poundes. Then take for the shillings and pence, as in the fifte rule of thys chapter is plainly declared. And as by these examples folowing may apere.

At	3. li.	6. shil.	8. d.
What	49.
	147.	.0	.
	16.	.6	.8
	163. li.	6. shil.	8. d.
At	5. li.	18. shil.	4. d.
What	543.
	2715.	.0.	.
	271.	.10.	.
	135.	.15.	.
	90.	.10.	.0
	3212. li.	15. shil.	0. d.
At	2. li.	7. shil.	4. d.
What	927.
	1854.	.0.	.
	185.	.8.	.
	154.	.10.	.
	2193. li	18. shil.	0.d.

8. So these rules do serue both to bye and sel, at such a price the elle, the yard, the pece, the poūd waight, or a∣ny other thing: how much such a thīg Lykewyse they are very necessary to cōuert al peces of gold and syluer in∣to poundes: for I may as wel say, at 4. shil. 8. d. the French crowne, what are 135. crownes worth?

9. When anye one of the summes (which is to be multiplied) is cōpoūd of many denominatiōs: & the other is of one figure alone: thē shal ye mul∣tiplye all the Denominations of the other summe, by the same one figure begīning first wyth that sūme which is least in value towardes your right hande, and bring the product of those pence into shillings, and the producte of the shillings into poūds, as by this example doth appeare

At 3. li. 9. shil. 8. d.

What 7.

24. li. 7. shil. 8. d.

10. But (if in any of y^e nūbers which are to be multiplied) ther be with it a broken number, you must (according to his denominator) take one or ma∣ny partes of the other nūber, as neede doth require: and set the nūber which cōmeth thereof, vnder the productes, adding thesame togither. As thus: At 5. li. 7. s. 8. d. the grosse, what shall. 34. grosse ½ cost? First

At 5. li. 7. sh. 8. d

What 34 1/7.

170. .0

11. .6. .8.

1. .14. .0

2. .13. .10

185. li. 14. shil. 6. d

you shall multiply 5. li. 7. sh. 8. d. by 34. grosse, saying 5. ty∣mes 34. doe make 170. li. then for 6. sh 8. d. take the ⅓ of 34 which is 11. li. 6. shil 8. d. Thirdely, for 1 sh. take 34. shillings, which is 1. li. 14. shillings 0.

Lastly, for the ½ grosse, you must take ½ of the 5. li. 7. s. 8. d. which is 2. li. 13. s. 10 And then adde them all togither, so you shall finde that the 34. grosse ½ at 5. pound 7. shillings 8. pence is worth

185. pound, 14. shillings 6. pence, as a∣peareth in the margent.

And as in this last exaumple, you did take the half of the money, (which one grosse was worth) for the ½ grosse Bycause that 1. grosse beeing worth 5. pound 7. shillings 8. pēce, the ½ grosse muste bee worth halfe so muche. So likewise, if you haue ⅓ of a grosse, or of any other thing, you muste take the ⅓ of the price, that one grosse is worthe. Semblably, for the ¼ of any thing you shal take the ¼ of the price, also if you haue ⅔, take the ⅔ of the price that one is worth, and of all other fractions, as by these examples folowing doth ap∣peare.

At 4. li. 6. shil. 8. d.

What 46. ½.

184. .0. .0

15. .6. .8

2. .3. .4

201. li. 10. 0.d.

At 8. li. 0. shil. 9. d.
What 54. ⅓
432. .0. .0
1. .7. .0
0. .13. .6
2. .13. .7
43. li. 14. shil. 1. d.
At 3. li. 16. shil. 8. d.
What 17. ¾
51. .0. .0
8. .10. .0
5. .13. .4
. .19. .2
68. li.. 00. shil. 10. d.

11. If you will make the proofe of these rules aforesayd, you must first abate the sūme of money (which the fractiō of the multiplicatiō doth import) frō the totall sūme. And diuide the rest of the poūds of the sayd total summe, by the whole multiplicand, the fraction only accepted. And if any thing do re∣main after the diuision is made, that remaine shal be multiplied by 20. and vnto the product of that multiplicati∣on,

you shall adde the shillings which remained of the rest of the total sūme Agayn, if any thing do remaine after the same diuision, you must multiply thesame by 12, & vnto the product adde the pence of the total summe that re∣mayned, if any be left. And thus if ye haue truly wrought, you shal find a∣gain the higher sūme of your questiō that is to say, the price that one grosse or any other thing is worth, whereof you demaund.

Or otherwise reduce the remayne of the totall summe (the value of the moneye that the fraction is worthe, beeing fyrste deducted) all into pence, in multiplying the pounds by 20, and the shillings by 12. adding thereunto, the shillings and pēce, which are ioy∣ned wyth the remayne of the sayd to∣tall sūme, if any such be, then diuide those pence by the foresayde number that is to be multiplied, the fraction of the same number beeing also aba∣ted. So shall you finde the price that

one peece, one Grosse, or any other thing is valued at. As in the firste ex∣ample going before, where the totall summe is 201. pound 10. shillings, frō the which I doe first abate the price of the half grosse, which is 2. li. 3. s. 4. d, the rest is 199. li. 6. s. 8. d. which beeing reduced into pens bringeth 47840. d. I diuide the same by 46. and thereof commeth 1040. pence. Then I diuide that 1040. pence by 12. and they bring 86. shillings 8. pence, that is to say, 4. li. 6. shillings eyght pence, whiche is the price that one grosse, or any other thing did cost, as in that fyrste exam∣ple doth appeare.

12. The lyke is to be done of any manner of thinge that is solde by the hundred or by the Kyntall.

As thus: at 12. pound 7. shillings. 6. d the 100. pounde wayghte: what shall 374. pound wayght cost? You shall fyrst multiply twelue pounde, seuen shillings, sixe pence, by three: that is to saye, by three hundreth. Then for

50. li. waight, you

At. 12. li. 7. sh. 6. d

What 3/74.

37. .2. .6

6. .3. .9

2. .9. .6

0. .9. .10. ⅘

46. li. 5. sh. 7. d. ⅘

shall take the ½ of 12. li. 7. s. 6. d. bi∣cause 50. li. is the ½ of 100. li. Like∣wise for 20. pound waight, which is the ⅕ of 100. li. 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, which will make the summe of 64. li. 5.s. 7. d. ⅘, as by this exāple aboue writ∣ten doth appeare.

The proofe is made by reducyng the totall summe into pence. And to diuide y^e product by the number y^t is to be multiplyed, y^t to to saye by 374. likewise diuide y^e quotient produced of that first diuisyon by 12. so shal you finde againe the higher summe 12. li. 7. shil. 6. d. which is the price of 100. li. waight, as before.

13. Also the like maye be done of

our vsuall waight here in Englande (which 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, which 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 hun∣dreth pounds waight, that is to saye, the 112. li. What shall 24. C. 3. quar. 21. li. cost after the rate?

Fyrst, you shall multiply 24. hun∣dreth by 3. which is the 3. li. & thereof cōmeth 72. li. then for 6. s. 8. d. which is the ⅓ of 20. s. you shal take y^e ⅓ of 24

which is 8. li. for

At 3. li. 6. sh. 8 d.
What 26. .3. 21. li.
72. .0. .0
8. .0. .0
1. .13. .4
. .16. .8
. .8. .4
. .4. .2
83. li. 2. sh. 6. d

24. nobles ma∣keth 8. li. after∣warde, for the 3. quarters of y^e C. you shal first for y^e 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. (which is the quar. of a C.) you shall take the ¼ of 3. li. 6. s. 8. d. or else the ½ of the product, which came of 56. li. which is 16. s. 8. d. likewise for 14. li. take the ⅛ of 3. li. 6. s. 8. d. which is 8. s. 4. d. or else the ½ of the product of 28. li. which is al one: lastly for 7. li take the 1/16 of 3. li. 6. s. 8. d. or else 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^e 24. c. 3. quar. 21. li. waight worth after 3. li. 6. s. 8. d. y^e C. as appereth in

the margent.

The proofe hereof is made, lyke to the other proofes aforesaide, sauing that wher in those proofes, you abate the price of the money, that the frac∣tion was worthe, from the totall summe: here in thys example (and in such 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 cō∣uert into pence, diuiding the product of y^e multiplication by the iust nūber of the hundrethes, so shall you finde the pence y^e one hundreth is worthe, which you shall bring into poundes by the order of diuision, & so all other.

¶The seconde Chapter treateth of the rule of three compounde, which are foure in number.

THere belongeth to the fyrste & seconde partes of the rule of thre compounde alwaies fyue numbers: wherof (in the fyrst

part of the rule of three compounde the seconde number and the fift, are alwaies of one semblaunce, and like denomination: whose rule is thus, multiply y^e first nūber by the seconde, & that shalbe your diuisor: then mul∣tiplye y^e other three nūbers the one by the other to be your diuidende. Exā∣ple, of this first part: if 100. crowns in 12. monthes, do gaine 16. li. what will 60. crownes gaine in 8. monthes? Aunswere, first multiplie 100. crownes by 12. monthes, & therof cōmeth 1200. for your diuisor: then multiply 15. li. by 60. crownes, & by 8. monthes, & you shall haue 7200. diuide 7200. by 1200. & therof cōmeth 6. li. so many li. wyll 6. crownes gaine in 8. monthes: thys questiō may be done by y^e double rule of 3. y^t is to say bi y^e rule of 3. at 2 times: yet this rule of 3 cōpoūd is more brief

Crownes.	monthes.	poūds.	crownes.	monthes.
100.	.12.	.15.	.60.	.8.
	1
	72	00
	12	00	(6. li.

2. In the seconde part of the rule of thre compound, the 3. number is like vnto the fift, wherof the rule is thus: multiplie the 3. number by the 4, the product shalbe your diuisor: thē mul∣tiply the first number by y^e seconde, & the product therof by the fift, y^e which number shal be your diuidend, or nū∣ber y^t is to be diuided: as by example.

When 60. crownes in 8. monthes do gaine 6. li. in how many monthes wil 100. crownes gain. 15. li. Aunswere: Multiply the thirde number 6. by the fourth nūber 100: & therof cōmeth 600 then multiplye y^e firste number 60. by the secōd nūber 8. & by y^e fift nūber 15. thereof will come 7200. then diuide 7200. by 600. & the quotiēt wilbe 12: in so many monthes will 100. crownes gaine 15. li. This question may lyke∣wise be done by the double rule of 3.

Crownes	monthes.	pounds.	crownes.	pounds.
60.	.8.		.6.	.100.	.15.
	1
	72	00	monthes.
	66	00	(12

3. In the thirde part of the rule of 3. compound, there may be 5. numbers or more: & in this rule y^e first nūber & the last are alwayes dissemblaunt y^e one to thother: & the questiō is from the last nūber vnto the first, wherof y^e rule is thus: multiply that number which you would know by those nū∣bers which do giue y^e value, & diuide y^e product of the same, by y^e multiply∣cation of the nūbers which are alrea∣dy valued, as by exāple. If 4. deniers Parisis, bee worthe 5. deniers Tour∣nois, & 10. deniers tournois, be worth 12. deniers of Sauoy, I demaūd how many deniers Parisis are 8. deniers of Sauoy worth? Aunswere: Multi∣ply 8. deniers of Sauoy (which is the nūber y^t you would knowe) by 4. de∣niers parisis, & by 10 deniers tournois which are the nūber y^t giue y^e value, & they make 320: then multiplie 5. de∣niers tournois, by 12 deniers of sauoy (which are y^e nūbers already valued) & thei make 60: lastly diuide 320. by 60

and you shal finde 5. deniers ⅓ parisis, so muche are the deniers of Sauoye worth.

Parisis.	tournois.	tournois.	sauoy.	sauoy.
4. d.	.5 d.	.10 d.	.12 d.	.8 d.
	32	0	par.
	6	0	(5. d. ⅓.

4. In the fourthe parte of the rule of thre compounde: the fyrst number and the last are always semblant and of one denomination, and the questiō of this rule, is alwayes from the last number to y^e last sauing one. Where∣of there is a rule which is thus. You must multiplye that number which you woulde knowe, by the numbers that are alreadye valued, and diuide the product of the same, by the multi∣plication which commeth of the nū∣bers that giue y^e value, as by exāple.

If 4. deniers Parisis, bee worth 5. Deniers Tournois, and 10. Deniers Tournois, be worthe 12. Deniers of Sauoy, I demaunde how many De∣niers

of Sauoy, are 15. Deniers Pa∣risis worth. Aunswere: Multiplye 15. Deniers Parisis that you woulde knowe, by 5. Deniers Tournois, & by 12. Deniers of Sauoye, which are the numbers alreadye valued, and thei make 900. Diuide the same by 4. times 10. which are the numbers that doe giue the value, and you shal finde 22. Deniers ½ of Sauoye, so much are the 15. Deniers Parisis worth.

Parisis.	tournois.	tournois.	Sauoy.	Parisis.
4. d.	.5 d.	.10 d.	.12 d.	.15 d.
	12
	90	0	Sauoy.
	44	0	(22. d. ½

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

IF 31. Deuonsh. dosēs do cost me 100. li. 15. shil. What shall 4. do∣sens cost? Aunswere: fyrst bring the 100. li. 15. shill. all into shillings,

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 di∣uide the product by 31. and the quoti∣ent wilbe 260. s. The which diuide a∣gaine by 20. and therof commeth 13. li.

Dosens.			Dosens.
31.	100. li.	15. sh.	4. d.
	20
	2015
	4
	8060.
	1
	28
	8060 (260.
	3111
	33

If foure Dosens be worth 13. pound. What are 31. Dosens worthe by the price? Aunswere: Multiply 31. by 13. and therof cōmeth 403. The which you shall diuide by 4. and thereof com∣meth

and thereof commeth 100. li. ¾, which ¾ are 15.s. and so much are 31. Dosens worth as before.

Dosens.	li.	Dosens.
4.	.13.	.31.
		13
		393
		1
		403.
	403
	444 (100. li. ¼

If 49. elles be worth 2. li. 4. s. 11. d. what are 18. elles worth by the price? First you must bring 2. li. 4. s. 11. d. all into pence, in multiplying 2. li. by 20. maketh 40. adde thereto 4. shil. they make 44. s. y^e which multiply by 12. d & thei make 528. d. whereunto adde 11. d all is 539. d. the which 539. d. muste be your second nūber in y^e rule of 3. then multiply 539. by 18. & therof commeth 9702. diuide y^e same by 49. & you shal haue in your quotient 198. d. y^e which diuide by 12. & you shal finde 16. s. 6. d. so much are the 18. elles worthe.

Elles:		Elles.
49.	.2.li. 4. sh. 11.d. .18
	20	539
	44	162
	12	54
	99	90
	44
	539	9702

13	1
427	76
386	198 (16. sh. 6.d
9702 (198.	122
4399	1
44

If 18. elles be worth 16.s. 6.d. what are 49. elles worth by y^t price? Auns. bring 16.s. 6.d. into pence, in multi∣plying 16. by 12. and thereof commeth 198.d. with the 6. d. added to it, then multiplye 198 by 49. the product will be 9702. The which diuide by 18. elles and therof commeth 539.d. Then di∣uide 539.d. by 12. and the product ther∣of by 20. So shall you haue 2. li. 4. sh.

11.d. so much are the 49. elles worth.

Elles.		Elles.
18.	.16. sh. 6. d.	.49
	12	198
	32	392
	166	441
	198	49
		9702

17	1
446	151
9702 (539.	••39 (44. shill.
1888	122
••1	1

If a yarde of Veluet cost 19.s. what shall ¾ of a yarde cost? Aunswere: sette down your numbers thus. If

1/1

19/1

¾

. Then multiplye 1. times 19. by 3. and therof cōmeth 57. for your diuidende, or number to be diuided. The which 57. you shall diuide by 1. times 1, foure times, which are 4, and your quotiēt wil be 14.s. ¼, which ¼ is worth 3.d. so

much are the ¾ of a yarde worth after 19. shil. the yarde, as by practise follo∣weth.

1/1	19/1	¾	11
			57 (14. sh. ¼
			44

Or otherwise by the rules of prac∣tise: first for 2/4 of a yarde which 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 nūbers togither, &

19. shil.

9. sh. 6.d.

4. .9.

14. .3.d.

you shall haue 14.s. 3.d. as aboue is said, and as appeareth here in the margent.

If ¾ of a yarde of Veluet do cost 14. shil. 3.d. What shall 1. yarde coste set your numbers downe thus: if

¾

14

¼

1/1

. Reduce 14. ¼ into a fraction, and they wil be 57/4 thē multiply 57. by 1.4. times, & thereof cōmeth 228. for your diuidend. Likewise multiply 1. times

4.3. times, & therof cōmeth 12. for your diuisor: then diuide 228. by 12. & your quotient wil be 19. shil. so much is the yarde of veluet worth.

	57		1
¾	14 ¼	1/1	10
			228 (19. shil.
			122
			1

Or otherwise by y^e rule of practise: you shall take the ½ parte of 14. sh. 3.d. and adde it with the same 14. sh. 3.d. and you shall haue 19. shill. as before.

14. shil.	3. d.
4.	.9. d.
19. shil.	.0. d.

If one ell of Hollande 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 diui∣dende: likewise multiply thre times 1. one time, thei make 3. for your diui∣sor, then diuide 10. by 3. & thereof com∣meth 30.s ½ which ⅓ is worth 4. pēce, &

so much are the ⅔ of an ell worth.

1/1	5/1	⅔	1
			10	(3. shil. ⅓
			3

Or otherwise, by the rule of prac∣tise: take first the ⅓ of 5.s. for the ⅓ of an ell, which is 1.s. 8.d. Likewise, for the other ⅓ of an ell take againe ano∣ther ⅓ 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.

5. shill.
1.	.8
1.	.8
3. shill.	4 d.

If ⅔ 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

⅔

3 ⅓

1/1

. First reduce 3 ⅓ all into thirdes, and it will be. 10/3. Then mul∣tiply 1. times 10.3. times, and thereof cōmeth 30. for your diuidēd. Likewise multiplie 1. times 3.2 times, youre quotiēt wil be 6. then diuide 30. by 6. & you shall haue 5. s. so much is the ell

of Hollande clothe worth.

	10		30
2	3 ⅓	1/1	6 (5. sh.

Or otherwise by practise, take the ½ of 3.s. 4.d. which is 1.s. 8.d. & adde it to the same 3.s. 4.d. and therof wyll come 5.s. as before. For the ⅓ of 5.s. is as much as the ½ of

3. sh.	4.
1.	8.
5. sh.	0.d.

3.s. 4. d. which was the price that the ⅔ of an elle did cost, as ap∣peareth.

If one ell cost me 17.s. what shal. 15. elles ⅛ part coste? which ⅛ is halfe a quarter of an elle. Aunswere: saye, of

1/1

17/1

15. ⅛.

. First reduce 15 ⅛ into eight partes, and they make 121/8 then mul∣tiplie 121. by 17.1 time, and thereof commeth 2057. for your diuidende. Likewise multiply 8. times 1. 1. time, and your quotient wyll bee 8. for your diuisor, then diuide 2057. by 8. and you shall fynde 257. sh. ⅛, whiche is 12. li. 17. shil. 1.d. ½ and so muche are

the 15 elles ⅛ worth, as by practise doth appeare.

		121
1/1	17/1	15 1/8

Or otherwise, for 10 sh. take y^e ½ of 15 which is 7 li. 10 sh. then for 5 sh. take the ½ of 7. li. 10.s. which is 3. li. 15.s. thirdly for 2.s. take y^e ⅕ of 7. li. 10.s. bi∣cause y^e ⅕ of 10. sh. is 2. sh. Fourthlye, for the ⅛ of y^e ell, you shal take the ⅛ of 17.s. which is 2.s. 1.d. ½. Lastlye, adde all these summes togither, and thē shal you find 12. li. 17.s. 1.d. ½ as before, and as appeareth more plainly in the margent.

15.	. ⅛.
17.s.
7.	.10.	.
3.	.15.	.
1.	.10.	.
	2.	1 ½.
12. li. 17.s. 1.d. ½

If 25. elles be worthe 2. li. 3.s. 4.d. what are. 18 elles ¾ worth by y^t price? Aunswere: first put 3.s. 4.d. into y^t part of a li. and you shal haue ⅙ then say, if 25/1 giue me 2. li. ⅙ what shall 18 ¾ giue: put y^e whole number into his brokē, and then multiplie 1. times 13. by 75. y^e product will be 975. the whiche you

shall diuide by 25. times 6.4. times, which maketh 600. Then diuide 975. by 600. and your quotient will be 1. li. and 375. remaineth, y^e which 375. you shall multiplie by 20. thereof cōmeth 7500. diuide y^e same by 600. your quo∣tient wil be 12.s. and 300. remaineth y^e which abbreuiated bringeth ½ which is 6.d: thus y^e 18 elles ¾ are worth 1. li. 12.s. 6.d. as by practise appeareth.

	13	75
25/1	2 ⅙	18 ¾.

Or otherwise by the rules of prac∣tise: for bicause that 12. elles ½ is the ½ of 25. elles, therfore take the ½ of 2. li. 3.s. 4.d. which is 1. li. 1.s. 8.d. then for 6. elles ¼ take ¼ of 2. li. 3.s. 4.d. or else the ½ of the last product (that is to say of 1. li. 1.s. 8.d.) which is all one, & adde them togither, so shal you haue 1. li. 12.s. 6.d. as before.

.2.	.3.	.4.
.1.	.1.	.8.
	.10.	.10.
1. li.	12.s.	6.d.

If 15. yardes be worth 32.s. what are halfe a yarde or halfe a quarter or else ⅝ of a yarde worth. Aunswere: say, if 15/1 giue 32/1 what wyll ⅝ giue? Mul∣tiplie 1 times 32. by 5. and diuide the product by 15. times 1. shil. and 4. re∣maineth, which is ⅓ of a shil. that is to say 4.d. and so much are the 5/2 of a yarde worth.

15/8

32/1

⅝

Or otherwise, se what the yarde is worth after the maner aforesaid in y^e other examples, & you shal find y^t the yard is worth 2.s. 1.d ⅗ of y^e which nū∣ber take first y^e ½ for 4/8 which is 1.s. 0 d. ⅘, of the which nūber, take the ¼ for y^e other ⅛ which is 3 d. ⅕, adde these two nūbers togither, and you shal finde y^e ⅝ to be worth 1.s. 4.d. as before is said

2. sh.	1.d.	⅗.
1.	0.	⅘.
1. sh.	4.d.	0.

If 13. els ⅚, be worth 27.s. what are 10. elles ⅔ worth by y^t price? Aunswere: say if 13. ⅚ giue 27/1, what shal 10. ⅔ giue: put the whole nūbers into their bro∣ken, & you shall finde 83/6, 27/1, & 32/3. Thē multiplie 6. times 27. by 32. & thereof commeth 5184. the which nūber you shall diuide by 83. times 1. thre times, and you shall finde 20. sh. 68/83 which is worth 9.d. 69/83 part of a penny.

83		32
13 ⅚	27/1	10 ⅔

If two yardes ½ be worth 4.s. 8.d. what are 8. yards ¼ worth? Aunswere: put the 8.d. into the part of a shilling, which wilbe ⅔ then reduce the whole numbers into their broken, and they will stande thus. 5/2, 14/3, 33/4, then multi∣plie two times 14. by 33. and diuide y^e product by 5. times 3.4. times, & you shall finde 15.s. 4.d. ⅘, so much are the eyght yardes ¼ worth.

5	14	33
2 ½	4 ⅖	8 ¼

If one kersey bee worth 2. li. 6.s. 8.d. how many kerseys shall I bie for 36. li. 3.s. 4.d. after y^t rate? Aunswere: put 6s. 8d. into the part of a li. & you shall haue 2. li. ⅓ for the first number in the rule of 3. and 1. ell for the second number: then put 3.s. 4.d. into the part of a li. and you shall finde 36. li. ⅙. for the third number, then will your 3. nūbers in the rule of 3. stande thus.

2. ⅓

1/1

36 ⅙.

. Therefore reduce the whole numbers into their broken, & you shal haue

7/3

1/1

217/6

. Then multi∣ply 3 times 1. by .217. & therof will cōe 651. for your diuidende. Likewise, multiply 7 times 1. by 6. & the product therof will be 42. Then diuide 651. by 42. and you shall finde 15. ½. So many kerseys of 2. li. 6.s. 8.d. the peece, shall you haue for 36. li. 3.s. 4. d.

7	217
2 1/7	36. ⅚

¶The 4. Chapter treateth of losses and gaines, in the trade of Marchaundise.

If 13. yards ⅓ be worth 22. li. 10.s. how shall I sel the yarde to gaine ⅓, or to make of 3.4? which is all one? Auns. say by the rule of 3. if 3. be come of 4. or if 3. yelde 4. what will 22. ½ yelde: multiplie & diuide and you shal fynde 30. li. Then say gaine by the rule of 3. if 13. yardes ⅓ doe giue 30. li. aswell of principal as of gaine: what wil 1. yard be worth by the price? Multiply and diuide, and you shall find 2. li. 5.s. and for that price must the yarde be solde to gaine the ⅓, or to make of 3.4.

		45	40.
5/1	4/1	22 ½	13 ⅓ 30/1	1/1

Or otherwise, take the ⅓ part of 22. li. 10.s. which is 7. li. 10.s. that shall you adde with 22. li. 10.s. and you shal haue 30. li. as be∣fore.

22.	.10. s.
7.	.10.
30.	.00.

Then di∣uide 30. by 13. ⅓, & you shall fynde 2. li. 5.s. as aboue is said.

If one yarde bee worth .27. sh. 6.d. for how much shal 16. yards ⅔ be solde to gaine 2.s. vpon y^e pound of money, y^t is to say: vpō 20.s. Answere, adde 2. vnto 20. and you shal haue 22, thē say: if 20.s. of principall, doe giue 22.s. as wel of principal as gaine: how much wil 27.s. 6.d. principall yelde. Multi∣plie and diuide & you shall finde 30.s. ¼: then saye againe by the rule of 3. if one yarde do giue me 30.s.¼ (which is aswel the principal as y^e gaine) what shall .16. yardes ⅔ giue 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. which is all one.

		55.		121	50
20/1	22/1	27 ½.	1/1	30 2/4	16 ⅔

If 10. yardes ⅔ be worthe 25. li. 10.s. For how much shal 2 yardes ¼ be sold to gaine after 10. li. vpon the 100. li. of money? Aunswere: say if 100. of princi∣pall

yelde 110. as well principall as gaine, how muche will 25. 10.s. yelde me? Multiply & diuide, and you shall finde 28. li. 1.s. Then saye if 10. yardes ⅔ do yelde me 28. li. 1. sh. aswel of prin∣cipal as of gaine, how much shal two yardes ¼ yelde me? multiply & diuide & you shall finde 5. li. 18.s. 4.d. 1/12, for so much shall the 2. yardes ¼ be solde to gaine after 10. li. vpō y^e 100. li. of mony.

		51
100/1	110/1	25 ½	10 ⅔	28 1/20	2 ¼

And although that in these questi∣ons of gaine and losse, sometimes the fyrst number is not like vnto y^e thirde number, that is to saye, of the same denomination: as one woulde saye: if 20.s. gaine 2. shil. what shall 50. li. gaine? or 25. li. &c. Or if 20. li. do gain 2. li. What shall 25.s. gaine mee, or what shall 27. sh. ½ gaine? Yet neuer∣thelesse, the rule is not therfore false. For if 20.s. doe gaine 2.s: 20. li. shall gaine 2. li. & 20.d. shall gaine 2.d. like∣wise

20. crownes shall gaine 2. crow∣nes, and so of all other: therefore it is to be vnderstand, that the first nūber in these reasons is presupposed to bee semblable to the thirde.

When one Marchant selleth wa∣res to another, and hee giueth to the byer 2. vpon 15: howe much shall the byer gaine vpō the 100. after the rate? Aunswere: say if 15. giue 17. what shall 100. giue? Multiplie and diuide, and you shall finde 113 ⅓, so the byer get∣teth after the rate of 13 ⅓ vpon the 100.

15

17

100.

If one northen dosen cost me 3. li. 5. s. & I sel the same againe for 3. li. 12.s. 6.d. how much doe I gaine vpon the pounde of money after y^t rate? Auns. say if 3. li. ¼ doe giue 3. li. ⅝ what shal 20/8 giue, put the whole nūber into their broken & you shal haue 13/4, 29/8, 20/1, then multiply 4. times 29. by 20. & thereof commeth 2320. for your number that is to be diuided, likewise multiply 13. times 8. 1 time, & therof cōmeth 104.

Then diuide 2320. by 104. & you shall finde 22.s. 4/13. So I shal get 2.s. 4/13 vp∣on 20.s. or vpon the li. of money.

13	29
3 ¼	3 ⅝	20/1

If a yarde of cloth cost me 7.s. 8.d. & afterwarde I deliuer out 13. yardes ¼, for 4. li. 13.s. 4.d. I woulde knowe whether I doe winne or lose, & how much vpon the 100. li. of money?

Aunswere: see first at 7.s. 8.d. the yard, what the 13. yardes ¼ shall coste, and you shal finde 5. li. 1.s. 7.d. And I sold them but for 4. li. 13.s. 4.d. so y^t I doe lose vpon the 13. yardes ¼ the summe of 8.s. 3.d. Then for to knowe howe much is lost vpon the 100: saye by the rule of three, if 5. li. 1.s. 7.d. doe lose 8.s. 3.d. What will 100. lose? Fyrst, put 1. shil. 7.d. into the part of a li. and it will be 19/240. Likewise put 8.s. 3.d. into the part of a li. and it is 33/80. Then will your nūbers stand thus 5 19/240, 33/80, 100/1, put the whole into his brokē, and

then multiply and diuide, so you shal finde 8. li. 1184/9752 which is worth .2. sh. 5. d. 169/1219 and so muche is loste vpon the 100. li. of money.

	1219.
5	19/240.	38/80	100/1.

More, if 12. yardes ½ of scarlet bee solde for 30. li. 15.s. vpon the which is gained after the rate of 11 1/9 vpon the 100. I demaunde what the yarde dyd cost at the first. Aunswere: from 30. li. 15.s. substract his 1/10 part which is 3. li. 1.s.6.d. and there resteth 27. li. 13.s.6.d the which number multiplied by 2. bringeth 55. li. 7.s. of the which is 11. li one shilling and foure pence.

Then take againe the ⅕ of the saide 11 pounde 1. shil. 4. pence, which is 2. pounde 4. shillinges thre pence. 9/25. And so muche did the Elle cost at the fyrst pennye.

30. li.	15. sh.
3.	1. sh.	6.d.
27.	13.	6.
2.
55.	7.	0.
11.	1.	4.	⅘.
2.	4.	3.	9/25.

More, if 15. yardes ¾ of scarlet doe cost me 32. li. 13.s. 4.d. And I sell the yarde againe for 2. li. whether doe I winne or lose, and howe much vpon the pounde of money.

Aunswere: Loke what the 15. yardes ¾ are worth at 2. li. the yarde, and you shall finde y^t they are worth 31. li.10.s. But they did cost 32. li. 13.s. 4.d. so y^t there is lost vpō y^e whole 1. li.3.s.4.d. Then, to know how much is lost vpō the li. say by the rule of thre, if 32. li. ⅔ doe lose 1. li. ⅙: what will ⅛ lose? that is to say, what will 1. li. lose? reduce the whole nūbers into their brokē, & then multiplie & diuide, so shall you

finde 21/588. part of a li. Then multiplye 21. by 240. bicause so many pence are in a li. & diuide the product by 588. so shall you finde 8.d. 336/588 which being abbreuiated doe make 7/4, & thus you se y^t 8.d. 4/7 is lost vpon the li. of money.

98	7
32 ⅔.	1 ⅙	1/1

If 1. yarde of cloth of tissue be solde for 3. li. 15.s. whereupon is lost after y^e rate of 10.s. vpon the 100. I demaunde what 12. yardes ½ of y^e same tissue did cost? Aunswere: adde vnto 3. li. 15. hys owne 1/10 part, which is 7.s.6.d. and al amounteth to 4. li. 2.s. 6.d. then loke what the 12. yardes ½ wil amount vn∣to, after 4. li. 2.s. 6.d. & you shal finde that they will come to 51. li. 11.s. 3.d. so much did the 12. yardes ½ cost.

3. li. 15.s.	12.		½
7.s. 6.d.	4. li. 2.s. 6.d.
4. li. 2s. 6d.	48.	.00.	.0.
	1.	.10.	.0.
	2.	.01.	.3.
	51. li.	11.s.	3.d.

More, if I sell one wilshire white for 6. li. 12.s. whereupon I doe gaine after the rate of 2.s. vpon the li. of money, that is to saye, vpon 20.s. I demaunde what 11. peeces of the same whites did cost mee? Aunswere: abate from 6. li. 12.s. (which is 132.s.) hys 1/11 part, & thereof cōmeth 12.s. and there remaineth 120.s. or 6. li. Thē see at 6. li y^e cloth, what the 11. clothes are worth 66. li. so much did the 11. clothes coste.

132. sh.	11
12. sh.	6
120. sh.	66. li.

If I sell 10. elles ½ of Hollād for 22 s.6.d. whereupon I do lose after the rate of 2.s. vpon the li. of money. I demaunde what the ell did cost mee? Aunswere: say by y^e rule of 3. if 18. giue 20.s. what will 22.s.6.d. giue? Mul∣tiplie & diuide, & you shall fynde 25.s. Then diuide 25.s. by 10. ½, & therof cō∣meth 2s.4 d. 4/7: So much did y^e el cost.

11/1

20/1

22 ½

If I sell one clothe for 5 li. wher vp∣on I do lose after 10 vpon y^e 100, I de∣maunde how muche I should lose or gayne vpon the 100, in case I had solde the same for 5 li.10 shil. Aunswere: saye, if 90 yelde 100, howe muche wyl 5. li. giue? Multiplie & diuide, & you shall finde 5. li. 5/9: then say againe by y^e rule of thre, if 5.5/9 come to 5. ½, what wyll 100. come vnto? Multiplie & diuide, & you shal finde 99. li. which being aba∣ted from 100. there wyll remaine 1. li. and so much is lost vpon the 100.

90.

100.

5.

5 5/9

5 ½

100/1

¶The 5. Chap. treateth of leng∣thes & breadthes of tapistry, and other clothes.

IF a peece of tapistry be 5 elles ¾ longe, and 4 elles ⅔ in breadth, how mani elles square doth the same pece conteine? Aunswere: Mul∣tiplie the length by the breadth, that is to saye 5. ¾ by 4. ⅔, and thereof commeth 26. elles ⅚ so many elles

square doth y^e same peece conteine.

More, if a peece of Tapistrie doe conteine 32. ells square, and y^e same being in lēgth 6. elles ¼. I demaūde how many elles in breadth y^e same peece doth conteine. Ans. diuide 32. elles by 6 ¼ and thereof commeth 5. 3/25: So many elles dothe the same peece conteine in breadth.

More, a peece of clothe beyng 13. yardes ⅓ in length, and 5 quarters 1/2 in breadth, how many yardes of ⅔ & ½ broade will the same peece make? Answere: see what parte of a yarde, the 5/4 and ½ be, and you shall finde y^t they make 1 yarde ⅜. Thē multiply 13. yardes ⅓ by 1 yarde ⅜ and you shall haue 18. yardes ⅓ in square y^e which you must diuide by ⅔ & ½ y^t is to saye by ⅚, (bicause y^t ⅔, ½ being brought into 1 fraction maketh ⅚) & you shall finde 22. yards: So many yardes of ⅔ & ½ large doth y^e same peece conteine.

More, a marchant hath bought 4. yardes ⅔ of cloth being syxe quarters ½ broade to make him a gowne the which he will line thorowout, wyth black Say of three quarters of a yarde broad, I demaund how much Say he must bye? Answere: Multiply y^e lēgth of the cloth, by the breadth, that is to say 4 ⅔ by 1. ⅝, (which is the syx quar∣ters ½) and therof commeth 7. yardes 7/12, the which diuide by ¾ & you shall finde ten yardes 1/9. So many yardes of Say must he haue to line the same 4. yards ⅔ of cloth of 6. quart. ½ broad.

More, at 6.s. 8.d. the ell square, what shall a peece of tapistre cost me, which is fiue ells ½ long and 4. ells ¼ broade? Aunswere, multiply 5. ½ by 4. ¼ and therof cōmeth 23. ells ⅜ square: then say by the rule of three, if one ell square cost me 6.s. 8.d. what shal 23. ⅜ cost? Multiplie and diuide, and you shall finde 7. li. 15.s. 10.d. so much the saide peece of tapistrie did cost.

Or otherwise, by the rules of prac∣tise, take the ⅓ of 23. 3/8: and you shall finde 7. li. 15.s. 10.d. as aboue is saide.

More, a peece of Hollande clothe conteining 42. elles ⅔ flemishe, how many elles englishe doe they make? Here must you fyrst note that 100. els flemishe, doe make but 60. elles eng∣lishe, and so consequentlye fiue elles flemishe doe make but 3. els english. Therefore say by the rule of 3. if 5. els flemishe doe make three ells english, how many elles englishe will 42. els ⅔ flemishe make. Multiplye & diuide, so shall you finde 25. elles ⅗ englishe, and so many elles englishe doth 42. 2/3 flemishe conteine, the like is to bee done of all others.

More, I haue bought a peece of Tapistrie, being 5. elles ¾ longe, and 4. elles ⅔ broade measure of Flaun∣ders, I demaunde howe many elles square it maketh Englishe measure?

Aunswere.

First, forasmuch as three ells english are worth 5 elles flemishe, therefore put 3 elles english into hys square, in multiplying 3. by him self which ma∣keth 9: likewise multiplye 5. in hym selfe squarely, and it wilbe 25. Then multiplye 5 ¾ which is the length of the peece, by 4 2/3 which is the breadth, & therof cōmeth 26 elles ⅚ square: thē say by y^e rule of three, if 25 elles square of flemishe measure, be worth 9 elles square of englishe measure, what are 26 elles flemish ⅚ worth? multiplie & diuide, and you shall finde y^t they are worth nine elles 33/80 square of english measure.

More at 3 s. 6 d. y^e ell flemish what is the englishe ell worth after y^t rate. Answere,: saye if 5. elles flemishe bee worth three ells english, what is 1 ell flemishe worth? multiply and diuide, & you shall fynde ⅗ of an englishe ell. Then saye by the rule of 3, if ⅗ of an englishe ell, be worth 3 s. 6 d. what is 1. englishe ell worth? multiplie and

diuide, and you shall finde 5 s. 10d. so much shall the englishe ell be worth.

More at 6 s. 8d. the flemishe ell square, what is y^e englishe ell worth. Answere, 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 ell: Then saye, if 9/25 of an englishe ell be worth 6 s. 8 d. what is one square ell englishe worth? mul∣tiplie and diuide, and you shall fynde 18 s. 6d. 2/9, so much shal one englishe ell square be worth.

¶The sixt Chapter treateth of y^e reducing of the paumes of Ge∣nes into english yardes, wher∣of foure Paumes maketh one englishe yarde.

I Haue bought 97. paumes ½ of Genes veluet, & I would know howe many yardes they wyll

make? Aunswere, Diuide 97. ½ by 4. and you shall haue 24. yardes ⅜. So many yards doe the 97. paumes ½ cō∣teine.

Or otherwise, take some other nū∣ber at your pleasure, as 20. paumes, which doe make fiue yardes, and thē 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 3/2 as before.

More, at two shillings 7.d. y^e paume 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 1/1 yarde worth? Multiplie & diuide, and you shall finde ten shil∣lings 4.d. So much is the englishe yarde worthe.

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

If 257. Paumes ½ bee worth 20. li. 16.s. 8.d. What is one yarde worthe after the rate? Aunswere, saye: by the rule of 3. if 257. ½ paumes be worth 20. ⅚, what are 4/1 paumes worth. Mul∣tiply and diuide, and you shall fynde 100/309 part of a pounde, which is worth 6.s. 5. pence, 5••/103: so much is one yard worthe.

¶The. vij. Chapter treateth of marchaundise solde by waight.

AT 9.d. ½ the ounce, what is y^e li. waight worth? Answere, say if 3/2 giue 9. ½ what will 16/1 giue multiply and diuide, & you shal finde 12.s. 8.d. so much is the yarde worth?

Or otherwise, by the rules of prac∣tise for syxe pence, take the ½ of 16. which is 8.s. then for 3.d. take the ¼ of 16.s. which is 4.s. Finally, for the halpenye, take 16. ob. which are 8.d. adde all these numbers togither and you shall finde 12.s. 8.d. as before.

More, at 10 d. ½ the ounce, what are 112. li. waight worth after the rate? Aunswere: reduce. 112. li. into oūces, in multiplying. 112. li. by 16. ounces & you shall haue 1792. ounces, thē say by the rule of 3. if

1/1

10 ½

1792/1

: Mul∣tiplie and diuide, and you shall finde 18816 d. which do make 78. li. 8 s. and so much are the 112. li. worth after 10.d ¼ the ounce.

At 12.s. 8d. the li. waight, what is the ounce worth? Answere: put 12.s. 8d. into pence, and you shall haue 152. pence: then say by the rule of 3. if 16. ounces cost 152d. what shall 1. ounce coste, multiplie and diuide, and you shall finde 9.d. ½, so much is the oūce 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. 2d. and you shall haue 9.d. ½ as before.

At 32. li. 10.s. the quintall, that is to

saye, the 100. li. waight: what is 1. li. waight worthe after the same rate? Aunswere, Put 32. li. 10.s. all into shil∣lings and you shall haue 650.s.

Then say, by the rule of three, if

100

650

1.

multiply and diuide, and you shal finde, 6.s. 6.d. so much is the li. worthe.

If one pound waight of saffron do cost me 18.s. 8.d. what shal 355. li. 10. oū¦ces cost me by y^e. same price? Aunswere saye by the rule of 3. if

1/1

18 ⅔

355 ⅝

. Multiply and diuide, & you shal finde 331. li. 18.s. 4.d. so much are the 355. li. ten ounces worth.

Briefe rules of vvaight.

WHo that multiplieth the pence that 1. li. waight is worth by 5. and diuideth y^e product therof by. 12. hee shal finde how many poūds in money the quintall is worth, that is to say, how much the 100. li. waight is worth.

And contrariwise he that multipli∣eth

the pounds of money that the 100. waight is worth by 12. and deuideth the product by 5. shall fynde how ma∣ny pence the poūde waight is worth.

¶Example.

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

More, at 13. li. the 100. li. waight, what is one pounde waight worthe? Aunswere, Multiplie 13. by 12. amoun∣teth to 156. the which diuide by 5. and you shal finde 31.d. ⅕ which is 2.s. 7.d. ⅕ and so much 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 hun∣dred.

Briefe Rules for measure.

Who that multiplieth the pence that one ell is worth, by 6. And diui∣deth the product by 12. hee shall fynde how many poundes in money y^e 120. elles are worth, which 120. elles wee count but for a C.

And contrariwise, hee that multi∣plieth the poundes in money that the 120. elles are worth by 12. and diuideth the multiplication 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 mo∣ney are 120. ells worth at 10.d. the ell.

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

The like is to be done of all ma∣ner

of wares, which are sold after 120. for the hundred.

¶Briefe Rules for our hundreth waight here at London, which 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 how many pounds in mo∣ney the 112. li. waight is worth.

ANd contrariwise, hee that multi∣plieth the poundes in money that 112. li. is worth by 60. and diuideth the product by 28. shall finde how many pence one li. waight is worth.

¶Example.

AT nine pence the pound waight, what is the 112. li. waight worth? Aunswere: multiplie 9.d. by 28, and thereof cōmeth 252, the which diuide by 60. & you shall finde 4. li. 12/60 which being abbreuiated is 1/5 of a pounde, which is worthe 4.s. And thus the 112. li. is worth 4. pound 4. shil.

At 8. li. y^e 112. li. waight, what is 1. li. waight worth? Answere, Multiplie 8. li. by 60. and thereof commeth 480, y^e which diuide by 28. & you shall finde 17.d. 1/7: so much is 1. li. waight worth.

¶The. viij. Chapter treateth of tares and allowances of mar∣chaundise solde by waight.

AT 12. li. the 100. suttell, what shall 987. li. suttell be worth? in giuing 4. li. waight vpon 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 12. li. what are 987. li. waight worth? multiply & diuide, & you shal finde 113. li. 23/26 which is worth 17.s. 8.d. 4/13. So much shal y^e 987. li. waight be worth.

104.

12.

987.

At 6.s. 8.d. y^e pound waight what shall 345. li. ½ be worth in giuing 4. li. waight vpon euery 100. for the tret. Answere, see first by the rule of three,

what the 100. pound is worth saying, if

1/1

6 ⅔

100/1

Multiplie and diuide, & you shal finde 33. li. ⅓ then adde 4. li. vnto 100. & they are 104. thē say againe by the rule of 3. if a 104. li. be solde for 33. li. ⅓ for how much shall 345. li. ½ be solde? multiply & diuide, and you shal finde 110. li. 14.s. 8.d. 12/13 So much shal the 345. li. ½ 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 reba∣ting 4. li. vpon euery 100. for Tare & Cloffe? Answere, Multiply 780. by 4. and therof commeth 3120. The which diuide by 100. and you shall haue 31. li. ⅕ abate 31. ⅓ from 780. and there wyll remaine 748 ⅘. 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. 18/25, and so much shall the 780. li. cost, in rebating 4. li. vpon euery 100. for Tare and Cloffe.

More, whether doth he lose more that giueth 5. li. vpō the 100. or he that rebateth 5. li. vpon the 100. for tare and cloffe? Answere. Fyrst, note that hee which giueth 5. li. vpon the 100. giueth 105. for 100: and he which rebateth 5. li. vpon the 100. giueth the 100. for. 95. Therefore say by the rule of 3. if 105. be giuen for 100. for how much shall y^e 100. be giuen? Multiply and diuide & you shal finde 95. 5/21: and he which re∣bateth 5. vpon the 100. maketh but 95. of 100: so that he loseth 5. vpon the 100. & the other which giueth 5. vpon y^e 100 loseth but 4. 16/21 vpon the 100. Thus he y^t rebateth 5. vpon the 100. loseth more by 5/21 vppon the 100. than the other which gaue 5. vpō the 100. for tare and cloffe.

If 100. of Allam doo cost mee. 26.s. 8.d. how shall I sell the li. waight to gaine after the rate of 10. vpon y^e 100. Answere, put 26.s. 8.d. al into pence, & you shal haue 320.d. Thē say by y^e rule of 3. if 100. giue 110. what shal 320. giue,

multiplie and diuide, and you shall finde 352.d. Thē say, if 100. li. be worth 352.d. what is 1. li. multiplye & diuide, and you shall haue 3.d. 26/50 which 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 how 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. ⅙ what will 100/1 yelde? Put the whole numbers into theyr broken, then multiplie and diuide, & you shal finde 104. 36/41 from the which substract 100. and there resteth 4. li. 36/41 so much 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

shall 100/1 giue? Put the whole num∣ber into their broken. Then multi∣plie and diuide & you shall finde 89. 1/16 the which you must substract frō 100. and there wyll remaine 10. li. 15/16, so much is loste vpon y^e 100. li. of money.

More if the li. waight doe cost mee 3.s. 2.d. & I sell it againe for 4.s. 4.d. how muche shall I gaine vpon 20.s. Answere: say if 3. ⅙ giue 4. ⅓ what shal 20/1 giue, Multiplie and diuide & you shall fynde 27.s. 7/19: out of the which abate 20.s. and there will remaine 7. shillings. 7/19, which is 4.d. 4/19: and so muche is gained vpon the pounde of money that is to say vpon 20.s.

If the pounde waight doe coste me 4.s. 4.d. and I sell it againe for 3.s. 2.d. I demaunde howe muche I shall lose vpon the pounde of money? that is to saye vpon twenty shillings.

Answere: say, if 4. ⅓ giue but 3. ⅙ what wil 20/1 giue, multiply & diuide & you

shal finde 14.s. 8/13 the which you must abate from 20.s. & there wil remaine 5.s. 5/13 which 5/13, is worth 4.d. 8/13 of a pennye and so much is loste vpon the pounde of money.

¶The. ix. Chapter treateth of certeine questions, done by the double rule, and also by y^e rule of three compounde.

WHen the quarter of wheate, doth cost 6.s. 8.d. the loafe of breade waying 20. ounces is solde for a ob. I demaund y^t if y^e quar∣ter of wheat did cost ten shillings, for how much shall the loafe of breade be solde, that wayeth 16. ounces?

Aunswere: by the fyrst part of y^e rule, of 3. compound which is mentioned in the thirde part of thys booke, and in the seconde Chapter of the same. Therfore say by the same first part of y^e rule of 3. cōpound, if

6. ⅔

20/1

½

10/1

16/1.

Then multiplye the fyrst number

by the seconde, and the product therof shalbe your diuisor. Likewise multi∣plie the other three numbers the one by the other, and the product thereof shalbe your diuidende: as thus, first multiply 6. ⅔ by 20/1, and thereof com∣meth 400/3 for your diuisor, then mul∣tiply ½ by 10/1 and the product therof by 16/1, so you shall haue 160/2 for your nū∣ber that is to be diuided, then diuide 160/2 by 400/3, and thereof commeth 480/800 y^e which being abbreuiated bringeth ⅗ of a peny: and for y^t price must the loafe of bread be solde, which wayeth 16. ounces, and the quarter of wheate being worth ten shillings.

Or otherwise by the rule of 3. at two times. First saye if 20/1 ounces giue, ½ what wyll 16/1 ounces gyue? Multiplie and diuide, and you shall fynde ⅖ of a pennye. Then saye a∣gaine, if 6. 2/7 doe giue mee ⅖, what will 10/1 giue? Multiplye and diuide, and you shall fynde ⅗ of a pennye, as afore is sayde.

When the cariage of one hundreth wayghte of marchaundise 50. miles doth cost 5s. what shall the cariage of 500 waight coste me for 16 miles?

Answere, By the fyrst part of the rule of 3 compound, saying if

100

50

5

500

16.

Multiply 100 by 50 the product wyl be 5000, which shal bee your diuisor. Then multiply 5 tymes 500 by 16 and therof commeth 40000 for your diui∣dend. Therfore diuide 40000 by 5000 and you shall finde 8 s. so muche shall coste the carriage of 500 wayghte 16 miles.

Or otherwise by the double rule of three, that is to saye by the rule of thre at two times: first say if 50 miles do paye 5 s. what shall 16 miles paye? Multiply and diuide, & you shall find 1 s. ⅗. Then say, agayne, if 100 waight do cost me 1 s. ⅗ what shal 500 wayght cost? Multiply and diuide, and you shall finde 8 s. as before.

When the cariage of 100. pounde waight of Marchaundise 84. miles, doth cost me six shillings, how many miles may I haue 64. pound waight caried for three. s. 4.d. Aunswere, by y^e second part of the rule of three com∣pounde: saye if

100/1

14/1

6/1

64/1

3 ⅓.

Then multiplie the fourth num∣ber 64/1 by the thyrde number 6/1, and thereof commeth 304/1 for your diuisor. Likewise multiplie 3 ⅓ by 100/1, and by 14/1 and you shall haue in the product 14000/3: then diuide 14000/3 by 384/1 and you shall fynde 72. miles 11/12 of a mile. So many miles shall y^e 64. li. waight be caried, for three shillings 4.d.

Otherwise by the rule of three, at two times: Fyrst say, if 100. waight doe cost me 6.s. what shall 64. pound waight cost? Multiplye and diuide, and you shall finde three shillings .21/25. Then saye, if 3. 21/25 bee payed for 84. miles cariage: for howe many miles shall 3.s. ⅓ be payed? Multiplye & di∣uide and you shall finde 72. miles. 11/12.

If 100. horses in 100. dayes do spende 180. quarters of otes: howe manye quarters of otes wil 350. horses spend in 150. dayes? Answere: by the fyrste part of the rule of three compounde: multiply 180. times 350. by 150. and di∣uide the product by 100. times 100: and you shall finde 945. quarters. So ma∣ny quarters of Otes will 350. horses spende in 150. dayes.

Or otherwise by the rule of 3. at two times: fyrst say, if 100. dayes doe yelde mee 180. quarters of otes: what shall 150. dayes yelde: multiplye and diuide, and you shall finde 270. quar∣ters: then say againe, if 100. horses do spende 270. quarters of Otes, howe many quarters of otes wyl 350. horses spend? Multiply and diuide, and you shall finde 945. quarters as before.

¶The tenth Chapter treateth of the rule of Fellowship, wyth∣out any time limited.

THe rule of felowship is thus: you must set down eche mās summe of money that he lai∣eth into company,* 1.12 euery one directly vnder the other, y^e which you shall adde altogither, & the totall sum of all their whole stocke beyng thus assembled, shalbe your common diui∣sor, to the finding out of euery mans part of y^e gaine. Then shall you mul∣tiplye the gaine, or the losse, by eche mans portion of money that he layde in, & diuide the products by the sayde diuisor: so shal you haue in your quo∣tient euery mans part of the gaine, or else of the losse, if any thing be lost.

¶Example

1. Twoo Marchaunts haue made companye togither, the first laide in 500. li. The seconde put in 300. li. and w^t occupying thei haue gained 64. li. I demaunde how much eche mā shal haue of the same gaines according to the money that he laide in. Aunswere: Adde 500. & 300. both togither, which

are the percels that they laide in, and therof commeth 800. for your diuisor: then say by the rule of three, if 800. li. (which is their stock) do gaine 64. li. what shal 500. li. gaine? (which is the fyrst mans money that hee laied in) multiplie & diuide and you shall finde 40. li. for the firste mans parte of the gaine: then say if 800. giue 64. what will 300. giue? Multiplye and diuide, and you shal finde 24. li. for y^e seconde mans part of the gaine.

500
300	800	64	500.
800
	800	64	300.

Or otherwise, put 500. li. which is the fyrst mans money y^t hee layed in, ouer the 800. li. which is the whole, stocke, and you shall haue 500/800 which being abbreuiated, do make ⅝, & such part of the gaine shal y^e fyrst mā take, y^t is to say ⅝ of 64. li. which is 40. li.

And consequentlye, by the same ma∣ner, the seconde shal take the ⅜ of 64. which is 24. pound for his part of the gaine as before.

5	00	3	00
8	00	8	00

2. Twoo Marchaunts haue com∣panied togither, y^e fyrst put in 640. li. and he taketh ⅝ partes of the gaine. I demaunde what the seconde Mar∣chaunt layed in? Aunswere, Seing that the fyrst Marchaunt taketh ⅝ of the gaine, it followeth that y^e seconde must haue ⅜ which is the rest, & ther∣fore say by the rule of three, if ⅝ of the gaine, which the fyrste man taketh, did lay into the stock 640/1. How much shall the ⅜ of the gaine laye in, which is the seconde mans gaine? Multi∣ply and diuide, & you shall find 384. li. so much ought the second man to lay into company.

3. Twoo Marchauntes haue com∣panied

togither, the fyrst man layed in 640. li. and y^e seconde hath layed in so much, that he must haue 60. li. for his part of 100. li. which thei haue gai∣ned. I demaunde howe much the se∣conde man did laye into companye? Aunswere: seing that the second man taketh 60. li. of the gaine, it followeth y^t the fyrst must haue but 40. pounde. Therefore say by the rule of three, if 40. li. do lay in 640. li. what shal 60. li lay in? Multiply and diuide, and you shall finde 960. pounde, so much did the seconde marchaunt lay in.

4. Two marchaunts haue com∣panied togither, the first laide in 83. li. 6.s. 8.d. y^e seconde put in 170. duckets: & thei haue gained 100. li. of the which the fyrst man muste haue 60. li. I de∣maund what the ducket was worth? Answere, seing that the first mā must haue 60. li. it followeth y^t the seconde must haue 40. li. therefore say by the rule of thre if 60. li. of gaine y^t the first

man taketh did lay in 83. li. 6.s. 8.d. of principall, howe much shall 40. li. of gaine put in, multiplye & diuide, and shall find 55. li. 5/9: so much are the 170. duckets worth. Then put 55. li. 5/9 into shillings, and you shall haue 1111.s. 1/9 thē to know what y^e ducket is worth, saye by the rule of three, if a 170/1 gyue 1111. 1/9, what will 1/1 giue? Multiplye and diuide, & you shall fynde 6.s. 6.d. 22/51, so much is the ducket worth.

5. Two Marchauntes haue com∣panied togither, the seconde mā laide in more by 30. li. than did the first mā: and they gained 120. li. of the which y^e first man ought to haue 50. li. I de∣maund what eche of them did lay in. Answere, from 120. li. abate 50. li. and there resteth 70. li. for y^e second mans part: so that by this meanes y^e seconde mā (bicause he laide in 30. li. more thā the first man did) taketh 20. li. more of y^e gaine: & therfore say by y^e rule of 3. if 20. li. of gaine did lay in 30. li. of prin∣cipall, how much shall 50. li. lay in?

Multiplye and diuide, and you shall finde 75. li. so much did the firste man lay in, and consequentlye the seconde layd in 105 li.

6. Two marchaunts haue compa∣nied togither, the second hath layd in twise so muche as the firste man dyd, and 10 li. more: and they gayned 100 li. of the which, the firste ought to haue 32 li. for his part: I demaūd how much eche of them dyd lay into company?

Answere, If it were not for the 10 li. that the second man layd in more: he should haue had but 64 li. of the gain which is the double of the first mans parte. But bicause he layd in 10 li. more, hee hath foure pounde more of the gayne, and therefore saye by the rule of three, if 4 li. of gayne did laye in 10 li. of principall, (which was ouer and aboue the double of the first mā∣nes laying in) what shall 32 li. of gay∣nes lay in? which is the firste mannes parte of the gaynes that hee taketh.

Multiplye and diuide, and you shall finde 80 li. for the first mannes laying in: and consequently 170 li. for the se∣cond mans portion that he layed in.

7. Two marchaunts haue compa∣nied togither, and they haue gayned 100 li. of the which the first must haue after the rate of 10 vpon the 100 li. and the second must haue after the rate of 15 li. vpon the 100 li. I demaunde how muche eche of them oughte to haue? Aunswere, Put 10 li. for the fyrst mans laying in, and 15 li. for the second mā∣nes laying in. Adde 10 li. and 15 li. togi∣ther, and they make 25 li. Then put 10 ouer 25. and it is 10/25 which being abbre¦uiated are ⅖. Therfore he that taketh 10 li. vpon the 100 li. must haue the ⅖ of the gayne, which is 40 li. Then put 15 ouer 25. and it is 15/25 which being ab∣breuiated are ⅕. Therefore the second must haue ⅗ of the 100 li. which is 60 li.

8. Twoo Marchauntes haue com∣panied

togither, y^e fyrst laide in 46. li. 18.s. and the seconde laide in 33. li.2.s. so they haue gained 30. li. I demaund how much euery man shall haue for his part of the gaine? Answere: Adde 46. li. 18.s. and 33. li.2.s. both togither and you shall finde 80. li. for your cō∣mon diuisor: then say if 80. li. which is all their stocke do gaine 30. li. what will 46.9/10 gaine, which is the fyrste mans laying in: Multiplie & diuide, and you shall finde 17. li. 11.s. 9.d. for the first mans part of the gaine. Thē say again, if 80. li. do gaine 30. li. what will 33. li. 1/10 gaine, which was the se∣conde mans, laying in: multiply and diuide, and you shall fynde 12. li. 8.s. 3.d. for the seconde mans part of the gaine.

And after the same maner shall you doe, in case there were three or foure Marchaunts that would com∣panye togither: Adding all theyre summes of money (which they laye into the stock) into one total summe:

which shalbe your common Diuisor: and then worke with the rest, as is taught in the former Questions of y^e rule of companie.

9. Three Marchaunts haue com∣panied togither, the first laide in I know not how much: the seconde did put in 20. peeces of cloth, and y^e thirde hath layde 500. pounde. So at y^e ende of their cōpany, their gaines amoun∣ted vnto a thousand pounde, wherof y^e fyrst man ought to haue 350. pound, and the seconde must haue foure hū∣dred pounde.

Now I demaunde how much the first mā did lay in, and for how much the 20. peeces of clothe were put into company?

Aunswere.

Seing that the fyrste and the se∣conde marchaunts must haue 750. li. for their parts of the gaine. Then the thirde man must haue the rest of the thousande pound which is 250. li.

And therefore say by the rule of thre, if 250. of gaine, become of 500. li. of principall: of howe much shall come 350. li. of gaine? which the fyrst man taketh, multiplye and diuide and you shall finde 700. li. So muche did the first man laye in: then say if 250. li. of gaine be come of 500. li. principall, of howe much will come 400. li. which is the gaine y^t the seconde mā taketh. Multiply and diuide, & you shal finde 800. li. For so much were y^e 20. peeces of cloth layde into company.

10. Three Marchaunts haue gai∣ned 100. li. the fyrst muste haue the ½, the seconde must haue ⅓: And y^e third must haue ¼. I demaund how much euerye man must haue of the gaine? Aunswere, Reduce ½, ⅓, ¼, into a cōmon denominatiō, after the order of the second reduction in fractions, & you shall finde 12/24, for the ½: 8/24, for the ⅓: and 6/24, for the ¼: Then take 12 for y^e first mans laying in, 8. for y^e se∣cond

mannes laying in: and 6 for the thirde mannes laying in. The which three numbers being added together shall be your common diuisor, which do make 26. 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 finde 46 li. 2/13 for the fyrste mannes part of the gaine. 30. li. 10/13 for the second mannes parte: and 23 li. 1/13, for the third mannes parte.

11. Two marchaunts haue gayned 100 li. the firste muste haue ½ and 5 li. more: the second must haue ⅓ and 4 li more: I demaunde how muche eche of them shall haue? Aunswere, From 100 abate 5 and 4. so ther wil remayn 91. then take the ½ of 100. li. which is 50 li. for the first mans laying in: Like∣wyse, 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 com∣mon

diuisor, then multiply 91. pound by 50. and diuide by 83. ⅓: and thereof cōmeth 54. pound, ⅗ vnto the which number adde 5, and all is 59. li. ⅗ for y^e first mans part. Likewyse multiplye. 91. by 33. ⅓: and diuide by 83. ⅓, & you shal finde 36. li. ⅖ vnto the which adde 4: and you shal haue fourty pound, ⅖ for the seconde mans part.

12. Twoo Marchauntes haue gai∣ned a hundred pound, the first muste haue the ½ lesse by 4. poūd, the second must haue ⅓: lesse by 2. pounde. I de∣maund how much eche of them shall haue? Aunswere, Adde 4. & 2. w^t 100. & they make 106. Then take as before is saide 50. pounde, for the first man, & 33. ⅓ for the seconde, adde them bothe togither, and they be 83. which shalbe your diuisor. Then multiplie 106. by 50. and diuide the product by 83. ⅓, so thereof commeth 63. li. ⅗. From the which abate the foure pounde lesse y^t the fyrst mā taketh, and then is there remaining 59. pound, ⅗ for hys parte.

Likewise multiplie 106. by 33. ⅓ and diuide by 83 ⅓ & 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 lai∣eth in, must be multiplied by the time that it remaineth in company: and of that which commeth therof you shal make their new layings in for eche of them: and then multiplye the gaines by euery one of them seuerally, the which you shall diuide by all their new layings in added togither, and you shall haue proporcionally eche mans part of the gaine according to his laying in.

¶Example.

1. Two Marchaunts haue compa∣nied togither, the first hath put in the fyrst of Ianuary 450. pounde, the se∣cond did lay in y^e 2. of May. 750. pound

And at the yeres ende, they had gay∣ned 100 li. I demaunde howe muche eche of them shall haue of the gayne? Answere: forasmuche as the firste dyd put 450 li. the fyrste of Ianuary: hys money remained in company 12. mo∣nethes, and therefore multiply 450. by 12 monethes, and therof commeth 5400. for his newe laying in. And the seconde layed in his 750 li. but at the first daye of Maye: so that his money remayned in companye but 8 mone∣thes. Therefore multiplye his 750 li. by 8. and therof commeth 6000 for hys new laying in: Then adde 5400. with 6000. and they make 11400 for youre common diuisor: Then multiply 100 li. which is the gaynes by 5400, and diuide the product by 11400. and ther∣of commeth 48 li. 7/19 for the first man∣nes part of the gayne. Likewise mul∣tiplye 100. by 6000, and diuide the pro∣ducte by 11400. and you shal finde 52 12/19 & so much must the second man haue for his parte of the gayne.

2. Two marchaunts haue compa∣nied togither, the first hath put in the first of Ianuary 640. li. The seconde can lay in nothing vntil the first of A∣prill. I demaunde how much he shall then laye in, to the ende that he maye take halfe the gaynes? Answere, Mul∣tiply 640 li. by 12. monethes that his money abideth in the companye, and therof cōmeth 7680 li. for his laying in. And so muche oughte the seconde mannes laying in to be, for bycause he taketh ½ of the gaine: But for that, that he putteth in nothing vntill the first of Aprill, his money can be in cō∣pany no lōger than 9 monethes. And therefore diuide 8680 by 9, and ther∣of commeth 753 li. ⅓ So much ought the seconde marchaunt to laye in the first of Aprill, to the ende that he may take the one moyty of the gaynes.

3. Three Marchauntes haue com∣panied togither, the firste layed in the firste of Marche 100 li. The se∣conde

laide in y^e first of Iune so much money, that of the gaine, hee must haue the ⅓ parte: and the thirde laide in y^e fyrst of Nouember so much mo∣ney, that of the gaines he must haue likewise ⅓ and thei continued in com∣pany, vntil y^e next Marche folowing. I demaunde howe much the seconde and the thirde Marchaunts did laye in? Answere, Multiply 100. which the firste man did lay in, by 12. monethes that his money continued in compa∣nie, and therof commeth 1200. for hys laying in: and so much ought the se∣conde and the thirde marchaunt eche of them to lay in: Bicause they parte the gaynes by thyrdes. But for that, that the seconde Marchaunt putteth in nothing tyll the fyrst of Iune, hys money can bee in companye but nine monethes. Therefore diuide 1200. by nine monethes, and therof commeth 133. ⅓. And so much ought the seconde Marchaunt to laye in: Then, foras∣much as the thirde Marchaunt, dyd

laye in nothing vntil the fyrst of No∣uember: His money abideth in com∣panye but the space of foure mone∣thes. Therefore diuide 1200. by 4. and thereof commeth three hundred pounde. And so much ought the third marchaunt to lay into companye.

4. Three marchauntes haue com∣panied togither, the fyrste layde in the fyrst of Ianuary a hundred Duc∣kettes. The seconde hath layed in fyftie pounde, the fyrste of Marche: And the thyrde put in a Iewell the fyrste of Iulye: And at the yeares ende, they had gained foure hundred crownes: of the which, the fyrste marchaunt must haue fifty crownes, and the seconde muste haue 80. I de∣maunde what y^e Ducket was worth, and at what price the Iewell was valued, whych the thyrde Marchaūt layde in?

Aunswere: the firste mannes money is 1200 as afore is sayde, and hee ta∣keth 50 crownes of the gayne: there∣fore say, if fifty crownes of gayne be come of 1200, which was his stock, of how muche shal 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 crow∣nes bee 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 lay∣ing in. Then diuide 1920, whiche is the seconde mannes laying in, by 10. monethes that his money did conti∣nue in company, and you shall finde 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: Fi∣nallye,

diuide 6480. (which is the third mannes laying in) by 6. mone∣thes that his Iewell remained in cō∣panye, and you shall finde 1080 Duc∣kets: and for that price was y^e Iewell put into company.

5. Three Marchauntes haue com∣panied togither: the first layed in the first of Ianuary 100 li. and the firste of Aprill 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 agayne 50 li. And at the yeres end, they found that they had gained 160 li. I demaunde how muche euery man shall haue? Answere, Multiply 100 li. which the first man layed, by 12 monethes, and therof commeth 1200. li. from that number abate 9 times 20 which are 180. and there wil remaine 1020. for the first mans laying in. Thē

multiplie 60. which the seconde man layde in, by ten & you shall haue 600. vnto the which adde 5. times one hū∣dred, which are 500. so all amounteth to 1100. for the second mans laying in: Afterwardes, multiplie 150. pounde, which the thirde man hath layed in, by 6. monethes, and therof commeth 900. from y^e which number abate thre times 50. and they are 150: so there resteth 750. for the thirde mans lay∣ing in. Then procede with the reste, as in the first Question of the rule of felowship with tyme, in adding 1020, 1100. and 750. altogither, which shall be your Diuisor: Then multiply 160 by 1020. by 1100. and by 750, & diuide at euery time by your Diuisor, which is by all theyr layings in added togi∣ther, and they make 2870, so you shall fynde 56. 248/287: for the fyrst man, 61. 93/287 for the seconde, and 41. 233/287 for the thirde man.

6. Two Marchaunts haue compa∣nied

togither, the firste hath put in 960. pounde, for the space of 12. mon∣thes, and he ought to haue 8. pounde vpon the hundred pounde of y^e gaine. The seconde hath layed in 1120. li. for y^e space of eight monethes, & he ought to haue after 12. pounde vpon the 100. pound of the gaine.

And at the yeres ende, they haue gained eyght hundred pounde. I de∣maunde how much eche of them shal 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 92160. for the fyrst mans laying in: then multiplye the 1120. that the second hath layed in by eyght mone∣thes, and that which commeth therof you shall multiplie againe by 12. and you shall finde 107520. for the seconde mans laying 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

finde 399 3/13 for the first man: and 430. li. 10/13 for the second man.

¶ The rule of company, be∣twene Marchaunts and their Factours.

7. The estimation of the bodye or persone of a Factour, is in suche pro∣portion to the stocke, which the Mar∣chaunt layeth in: as the gayne of the sayd Factour is vnto the gayne of the sayd Marchaunt. As thus: if a Mar∣chaunt do put into the handes of hys Factour 200 li. to employe, and he to haue halfe the profite, the persone of the sayd Factour shal be esteemed 200 li, And if the Factour do take but the ⅓ of the gaine, he should 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 Marchant layeth in, that is to say 100 li.

And if the Factour did take the ⅖ of the gayne, then the Marchaunt shall take the residue, which are ⅗ of y^e gain 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 per∣sone of the Factour, say if 3 giue me 2 what wil 200 giue? Multiplie 200 by 2 and diuide by 3 so you shall finde 133 ⅓ Otherwise, consider that the Fac∣toure taketh the ⅔ of that whiche the Marchaunte taketh. And therefore take the ⅔ of 200, and you shall fynde 133 ⅓ as before: and so much is the per∣sone of the Factoure esteemed to bee worth.

8. And if the Marchaunt should de∣liuer 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 gain: I demaund for how much shal his person be estemed Ans. abate 40 li. from 200 li. and ther

will remaine 160. li. And at so much shall his person be estemed.

And if the factour woulde take the ⅔ of the gaine, his person with his 40 pounde, shall bee estemed twise as much as the stock that the marchant layeth in, which shoulde haue but ⅓ of the gaine: for ⅔ vnto ⅓, is in double proportion. Therefore double two hundred pounde, therof cōmeth 400. li. from the which abate 40. li. & there will remaine 360. li. And if the Fac∣tour would take but the ⅓ of y^e gaine, that shall bee but the ½ of ⅔ which the marchaunt taketh: then the estima∣tion of his person, with his laying in should be estemed but the halfe of y^e which the marchaunt layeth in: take therfore the ½ of 200. li. which is 100. li. from the which abate fourty pounde, and the rest which is 60. li. is the esti∣mation of his person.

9. If it so chaunce that for to make traffick of 240. li. the person of y^e fac∣tour should be so estemed, y^t he shuld

haue but the ¼ of the gaine, & yet hee would haue the ⅔, I demaunde how much he shoulde put in of ready mo∣ney, besides hys person? Aunswere, seing that his person gaineth the ¼, al the whole laying in, shall gaine the rest that is to saye the ¾: nowe for bi∣cause ¼ is the ⅓ of ¾ therfore his person shalbe estemed the ⅓ of all the laying in. Take then the ⅓ of 240. and you shall haue 80. for the estimation of his person, and for that, that he wil 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 must lay in the ouerplus, which amounteth to 160. li.

10. A marchaunt hath deliuered to his Factour 1200. li. to gouerne them in the trade of Marchādise vpon such condition that hee for hys seruice

shal 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 gain, hys persone ought to bee estee∣med as muche as ½ of the stock which the Marchaunte layeth in, that is to say the ½ of 1200 li. which is 600 li. The reason is, bycause the ⅓ of the gayne that the Factoure taketh, is the ½ of the ⅔ of the gaine that the Marchaunt taketh.

11. A Marchaunt hath deliuered vn∣to his Factour 1200 li. and y^e Factour layeth in 500 li. and his person: Now, bicause he laieth in 500 li. and his per∣sone, it is agreed betwene them that he shal take the ⅖ of y^e gayn: I demaūd for how much his persone was estee∣med? Aunswere, Forasmuch as the Factour taketh the ⅖ of the gaine, he taketh the ⅔ of that which y^e Marchāt taketh, for ⅖ are the ⅔ of ⅗: and there∣fore

the Factours laying in ought to be 800. pounde, which is the ⅔ of 1200. pound, that the marchaunt layed in: Then abate 500. pounde, which the Factour did lay in, from 800. pounde, which should be hys whole stock and there remaineth thre hundred poūde for the estimation of hys person.

12. More, a marchaunt hath deliue∣red vnto his factour a thousand poūd vpon such conditiō, that the Factour for hys paines and seruice, shall haue the gaines of 200. pounde, as though he layde so much in of ready money: I demaunde what portion of y^e gain, the saide Factour shal take? Answere: See what parte the 200. li. (which the Factour layed in) is of 1200. which is the whole stocke of their company, & you shall finde that it is the ⅙, and such parte of the gaine shall the Fac∣tour take.

But in case, that in making the co∣uenauntes, it were agreed that the

Factour shoulde haue the gaine of two hundred pound of y^e stock, which the marchant layeth in, that is to say, of the thousand pound. Then should y^e Factour take the ⅕ part of y^e gaine. For 200. li. is the ⅕ of a 1000. pounde.

¶The xj. Chapter treateth of the Rules of barter.

[unspec I] TWoo Marchants wil chaūge their marchandise, the one w^t the other. The one of them hath cloth of 7.s. 1.d. y^e yarde to sell for ready money, but in barter he will sell it for 8.s. 4.d. The other hath Sinamon of 4.s. 7.d. y^e li. to sell for ready money. I demaund how he shall sell it in barter to the ende he be no loser? Answere, say, if 7. 1/12 (which is y^e price y^t the yard of cloth is worth in redy money) be solde in barter for 8. ⅓ for what shal 4.7/12 be solde in bar∣ter which 4. 7/12, is the price y^t the li. of Synamon is worth in ready money, reduce the whole numbers into their

brokē, and then multiply and diuide, and you shall finde 5.s. 4.d. 12/17 parts, of a peny, and for so much shall he sell the pounde of Synamon in barter.

2. Two Marchaunts wil chaunge their marchaundise the one with the other, the one of them hath Chaum∣lets of two pounde 18.s. 4.d. the peece to sell for ready money, and in barter he wyll sell the peece for 4. li. 3.s. 4.d. the other hath fine capps of 35.s. 10.d. y^e dossen to sell in barter. I demaund what y^e dossen of caps did cost in redy money? Answere, say if 4. li. 3.s. 4.d. which is the ouerprice of the peece of Chamlet, become of 2. li. 18.s. 4.d. which was the iust price of the same, of what shall come 35.s. 10.d. which is the ouerprice of the dossen of cappes? Multiply and diuide, & you shall finde 25.s. 1.d. and so much are the dossen of capps worth in redy money.

3. Two Marchaunts wil chaunge their marchaundise the one with the other: y^e one of them hath Fustians

of 18.s. 4.d. the pece to sell for readye money, and in barter hee will sell the peece for 26.s. 8.d. The other hath tapistry of 15.d. the ell to sell for readie money, and in barter hee wyll sell it for 20.d. the ell: I demaunde which of them gaineth, and how much vpō the hundred pounde of money?

Aunswere: saye if 18.s. ⅓ (which is the iust price of the peece of Fustian) bee solde in barter for 26.s. ⅔: for howe much shall 1.s. ¼ (which is the iuste price of the ell of tapistry) bee solde in barter? Multiplie and diuide, & you shal finde 21.d. 9/11. And he doth ouer∣sel it but for 20.d. so that of 21.d. 9/11: he maketh but 20.d. And therefore saye by the rule of three, if the seconde marchaunt, of 21 9/11, do make but 20/1 how much shall he lose vpon the 100/1? Multiplie and diuide, and you shall finde 91. ⅔, y^e which being abated frō a hūdred there wil remaine 8. ⅓. And after y^e rate of 8. ⅓. doth y^e secōde mar∣chāt lose vpō y^e 100. And consequētly,

the first marchaunt, of 20.d. maketh 21.d. 9/11: and therefore saye againe by y^e rule of three, if the first marchaunt of 20/11, do make 21.9/11 how much shall he gaine vpon 100/1? Multiplie and di∣uide, and you shall finde 109. li. 1/11.

Thus the fyrst gaineth after the rate of 9. li. 1/11: vpon the hundred pounde of money.

For your better vnderstanding of these Questions, you must note that when one marchaunt gaineth of an other after the rate of ten pound vpō y^e hundred pound he gaineth the 1/10 of his owne principall, and the other which loseth after the rate of 9. 1/11 vp∣on the hundred he loseth the 1/11 of his principal. And it may be proued thus: When one marchaunt will sell hys wares vnto another, which wares stande him but in 100. li. & hee will sell them for 110. li. he, of his 100. li. maketh 110. li. wherfore hee gaineth after 10. li. vpon the 100. which is the 1/10 of hys principall, and the other which byeth

wares for 110. li. that cost but 110. poūd of the 110. pound he maketh but 100. li. And therefore say by the rule of thre, if 110. become of 100. of howe much shall come 100? Multiplie and diuide, and you shall finde 90.10/11, the which abate from 100: and there resteth 9.1/11 is the 1/11 of hys principal that the se∣conde loseth vpon the 100. as afore is saide. And therefore, who so that wil know what one Marchāt gaineth of another, either after the rate of tenne vpon the hundred, which is the 1/10 of of hys principall, or else after the rate of twenty vpon the hundred which is the ⅕, or of any other parte, and that he would likewise knowe what part the other loseth of hys principall: hee must take for the numeratour of the broken number of hym that loseth, as much as for him that gaineth, thē adde the numerator and the denomi∣nator (of the broken number of hym that gaineth) both togither, & make thereof the denominator of the brokē

number of him y^t loseth, & then shall you haue the part of him that loseth, as by exaumple, of him that gayneth after ten. li. vpon the 100. li. which is the 1/10 of hys principall: take the nu∣meratour which is 1. and make that the numerator of the broken number of him that loseth, then adde 1. which is y^e numerator of the fraction of him that gaineth with ten, whych is hys denominator, & you shall haue 11. for the denominatour of the fraction of him that loseth. Then put one ouer the 11. and so you shal haue 1/11. Thus it appereth when one marchant gai∣neth of another after ten vppon the hūdred, he gaineth the 1/10 of his prin∣cipall, and the other loseth 9. 1/11 which is the 1/11 of his principall. And yf hee would gaine after 20. vpon the hun∣dred which is the ⅕ of hys principall, the other shoulde lose 16. ⅖ which is the ⅙ of hys principall, and so is to bee vnderstande of all other fractions.

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 sell for readye money, and in barter he will sell the peece for 23.s. 4.d. & yet hee will gaine moreouer after ten pounde vpon the hundred pound. The other hath woll of 50.s. the hundred to sell for readye money. I demaund how hee shall sell the C. of woll in barter? Aunswere, say if 20.s. 10.d. which is the iust price of the peece of Sey, be solde in barter for 23.s. 4.d. for how much shall 50.s. (which is y^e iust price of y^e C. of woll) be solde in barter? Multiplie & diuide, & you shal finde 56.s. Thē for bicause the first marchant gaineth after 10. li. vpon the C. li. he maketh of his C. li. 10. li. & consequently the seconde mar∣chaunt maketh of 110. li. but 100. li. And therfore say, if the seconde mar∣chaunt of 110. doe make but 100. how much shall he make of 56: Multiplie & diuide, & you shal find 50.s. 10.d. 10/11 of

a peny, and for so much shall he sell y^e hundred of woll in barter.

5. More, twoo Marchauntes wyll chaunge their marchaundise, the one with the other, the one of them hath Taffeta, of 16. crownes the peece to sell for redie money, and in barter he will sell the peece for twēty crownes, and yet he wyl gaine moreouer after ten pounde, vpon the hundred poūd. The other hath ginger of 3.s. 9.d. the pounde waight, to sel in barter. I de∣maunde what the pounde dyd coste in readye money? Aunswere: saye if twēty crownes which is the surprice of the peece of Taffata, become of 16. crownes the iuste price, of how much shall come. 3.s. 9.d. which is the price of the ouerselling the pound of Gin∣ger? Multiply & diuide, and you shall finde 3.s. Then, for bicause that the Marchaunt of Taffeta wil gayne af∣ter the rate of ten vpon the hundred: say if 100. doe giue 110. what shall 3.s. giue? Multiplye and diuide, and you

shall finde 3.s. 3.d. ⅗ and so much dyd the pounde of Gynger cost in readye money.

6. More, two marchaunts wyll chaunge their marchaundise the one with the other, the one of them hath Worsteds of 25.s. the peece to sell for ready money, and in barter, hee wyll sell the peece for 33.s. 4.d. and yet hee loseth after ten vpon the hundred: the other hath waxe of 3. li. 6.s. 8.d. the hundred to sell for readye money. I would know howe he should sell hys waxe in barter? Aunswere: say if 25.s. which is the iuste price of the peece of Worsted bee solde in barter for 33.s. 4.d. for how much shall three pounde 6.s. 8.d. be solde, which is the iuste price of the hundred of waxe. Multi∣plie and diuide, and you shall fynde 4. li. 4/9 which is 8.s. ten pence, ⅔ then for bycause that the Marchaunt of Worsteds, loseth after ten vpon the hundred: Of a hundred hee maketh

but 90. And therefore, saye: If 90. giue 100. what giueth 4. pounde. 4/9? Multiplie and diuide, and you shall fynde 4. 76/81 which is worthe 18.s. 9.d. 5/27, and for so much shall he sell y^e 100. of Ware in barter.

7. More, two Marchaunts wyll chaunge their marchaundise the one wyth the other, the one of them hath Worsteds of 5. pounde 6. shillinges, eight pence the peece to sell for ready money, and in barter he will sell the peece for 6. pounde, 13. shillings. 4.d. and yet he loseth after ten vppon the hundred, and the other hath Muske of two shillings, nine pence ⅓, the poūd waight, to sell in barter? I demaūde what the pound did cost in ready mo∣ney? Answere: say if 6. pound. ⅔ which is the ouerprice of the peece of Wor∣sted, become of 5. pound, ⅓ which is y^e iuste price of the same, of how much shall come two shillings 9. pence. ⅓.

Multiplie and diuide, & you shal finde 2. 2/9 which is 2.d. ⅔ then for bicause y^t the marchaunt of Worsteds loseth after ten vpon the hundred, of a hun∣dred he maketh but 90. and therefore say if 100. giue but 90. how much shal 2.s. 2/9 giue? Multiplie and diuide and you shall fynde 2.s. and so much cost the pound of Muske in ready money.

Other Rules of Barter, vvherein is giuen some part in ready money.

WHen a Marchaunt ouer sel∣leth hys marchaundise & he wyll giue also some part of hys ouerprice in ready mo∣ney as the ½ the ⅓ or the ¼ &c. He must substract the same parte of money frō the iuste price, and also from the ouer price of hys marchaundise: and the two numbers that remaine after the substraction is made, shalbe that two first numbers in the rule of three and the iuste pryce of the seconde mar∣chaunt

shalbe the third, to know how much he shall ouersell the part of his marchaundise.

8. Two Marchaunts wyll chaūge theyre marchaundise the one wyth the other, the one of them hath fyne woll at fiue pound the hundred, to sel for ready money, and in barter he wil sell it for six pounde, and yet hee wyll haue y^e ⅓ in ready money. The other hath cloth of 13.s. 4.d. to sel for ready money. I would know how hee shall sell y^e same in barter? Aunswere: take the ⅓ of 6. li. which is the ouerprice of the 100. of wolle, & you shall haue 2. li. the which abate frō 5. li. which is the iust price of y^e 100. of wolle & from 6. li. which is the ouerprice, and there shal rest 3. li. and 4. li. for the two first nū∣bers in the rule of three, thē take 13.s. 4.d. which is the iust price of a yarde of cloth for the thirde number: Then multiply and diuide & you shal finde 17.s. 9.d. ⅓: for so muche shall the se∣conde sell his cloth in barter.

9. More, twoo Marchauntes wyll chaunge their marchaundise the one with the other, the one of them hath waxe of thre pound 6.s. 8.d. the C. to sel for readie money, and in barter he will sell the same for 4. li. 3.s. 4.d. & yet he will haue the ¼ in redy money: and the other hath fine Crimson sat∣tine of 15.s. the yarde to sel in barter. I demaund what it is worth in redy money. Answere, Take the ¼ of 4. li. 3.s. 4.d. and abate it frō 4. li. 3.s.4.d. and from three pounde 6.s. 8. pence, and there resteth 3. li. 2.s.6.d. & 2. li. 5.s. 10.d. for the two first numbers in the rule of three, and 15.s. for y^e thirde number which is the ouerprice of the yarde of sattine. Then multiply and diuide, and you shall fynde 11.s. And so much did the yarde of Sattine cost in ready money.

10. Twoo Marchaunts will chaūge their marchaundise the one wyth the other, y^e one of them hath tinne of 50. shillings the hundred to sell for ready

money, and in barter he wil sell it for three pounde 6.s. 8.d. and hee wyll gaine after ten vpō the hundred, and yet he will haue y^e one halfe in ready money: and the other hath leade of 3. halfepence the li. to sel for redy mo∣ney. I demaund how hee shal sell the pounde in barter? Answere: See first at ten vppon the hundred, what the three pound ⅓ wil 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 whych he demaundeth in ready money, is 36. shillings and 8. pence, the whych being abated from fyftye shillings, and also from three pounds 13. shillings 4. pence, there shall reste 13. shillings 4. pence, and one pound 16.s. 8.d. for the twoo firste numbers in the rule of three, which you muste put al into halfepence, and thre halfe∣pence for the thirde number, and thē multiply and diuide, & you shall finde 4.d. ⅛, and for so much shall he sell y^e pounde of leade in barter.

11. More, twoo marchauntes wyll chaunge their marchaundise the one with the other, the one of them hath steele of 16.s.8.d. the hundred waight to sell for ready money, & in barter he will sell it for 25.s. and yet hee loseth after ten vpon the hundred, but hee wyll haue the ½ in readye money, the other hath yron of 6.s. 8.d. the hun∣dred to sell in barter, I demaunde what it did coste in readye money? Aunswere: say if a hundred come but to 90. how much shall 25.s. come to? Multiplie and diuide, and you shall fynde 22.s.6.d. of the which number, take the ½ which is 11.s. 3.d. & substract it from 22.s.6.d. and from 16.s.8. pēce and there shall rest 11.s.3.d. and 5.s.5. pence, for the two fyrst numbers in y^e rule of thre, and 6.s. 8.d. which is the ouerprice of a hundred of yron for the thyrde number, then multiplie and diuide, and you shal finde 3.s. 2. pēce, 14/27: & so much did the hundred of yron cost in ready money.

12. More, twoo marchaunts wyll chaunge their marchaundise, the one with the other, the one of them hath seyes of 20.s. 10.d. y^e peece to sel for re∣dy money, & in barter he wyll sell the peece for 21.s. & hee wyll haue the ¼ in ready money: The other hath capps of 35. shillings the dossen to sell for re∣dy money: but hee wyll gaine after ten vpon the hundred. I demaunde how he shall sell y^e same caps in bar∣ter? Aunswere: saye if a hundred bee worth 110. What shal 35.s. be worth, which is the iust price of the dossen of cappes? Multiplie and diuide, and you shall fynde 38. shillings 6. pence. Then take the ¼ of 25. which is 6.s. 3.d. and substract it from 20.s. 10.d. & from 25.s. and there shall rest 14.s. 7.d. and 18.s. 9.d. for the twoo fyrste numbers in the rule of three, and 38.s 6.d. which is the iust pryce wyth hys gaine of the dossen of cappes, for the thirde number: then multiplie and diuide, and you shall finde 49.s. 6.d.

and for so much hee shall sell the dos∣sen of caps in barter.

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

FIrst, you must note, that at Andwerpe they vse to make their accomptes by Deniers de gros, that is to saye by pence Fle∣mishe, whereof 12. doe make 1.s. Fle∣mishe, and 20, shillings Flemishe doe make 1. li. de gros.

1. If I deliuer in Flaunders, 500. li. Flemishe, at 19.s.6. de gros that is to say at 19.s. 6.d. Flemishe, to receaue 20.s. at London, I demaunde how much I shal receaue sterling at Lon∣don for the sayde 5. hundred pounde Flemishe? Answere,: Say, if 19 ½ giue 20/1, what will 500/1 giue? Multiplie & diuide, and you shal fynde 512. li.16.s. 4. pence 12/13 of a pennye. And so much sterling shal I receaue in London for

my 500. li. Flemishe.

2. If I deliuer in Londō 375. li.ster∣ling, to receaue in Andwerp 21.s.9.d. de gros, that is to say Flemishe, for euery pounde sterling. I demaunde how many poundes Flemish I shall receaue in Andwerpe, for the sayde 375. li. sterling? Answere, say if 20/1 giue 21. ¾: what will 375/1 giue? Multiplie & diuide, and you shall finde 407. li. So many poūds Flemish shal I receaue for the said 375. li. ster. in Andwerpe.

3. If I take vp money at Andwerp after 19.s. 6.d. flemishe to pay for the same at London 20.s. ster. and when the day of paiment is come, I am for∣ced to rechaūge the same, and to take vp money againe here in London to repaye the same, so that for twentye shillings, which I take vp here, I must repay, 19. shillings 9 d. at And∣werpe. I demaunde whether I doe winne or lose, and how much vpon y^e 100. li. of money? Answere, Say. if 19. 3/4 giue 19. ½, what will 100/1 giue?

Multiplie and diuide, and you shall finde 98. 58/79, the which beyng abated frō a hundred there wil remaine 1. 21/79. And so much doe I lose vpon the 100. pounde of money.

4. If I take vp at London 20. shill. sterling to pay at. Andwerpe 21. s. 8. d. Flemishe, and when the day of pay∣ment is come. I am constrained to take vp money againe at Andwerpe wherwith to repay the foresaide sum: and there I doe receaue 22. shillings. Flemishe to pay 20. shill. at London. Nowe I demaunde whether I doe winne or lose and howe much vpon the 100. li. of money after the rate? Answere, say if 21. ⅔ giue 22/1. What wil 1000/1 giue? Multiply & diuide, and you shall finde 101. 7/13, frō the which abate 100. and there wyll remaine 1. 7/13, and so much shal I gaine vpon the 100. li. of money.

The exchaunge from London in∣to Fraunce, is not lyke as it is

into Flaunders but is deliuered by the French crowne, which is worth 50. souse Tournois the peece.

And here muste you note that in Fraunce they make theyr accompte by Deniers Tournois, wherof 12. ma¦keth one souse Tournois, and 20. sou. Tournois maketh 1. li. Tournois, which they cal a Liuer, and y^e Frēche Crowne is currant amonge Mar∣chaunts for 51. souse Tournois, but by exchaunge it is otherwise, for thei wyll deliuer but 50. sou. Tourneys, which is. 2. li. 10. sou. Tournois for a crowne, or at such price as the takers vp of money can agree wyth the deli∣uerer. As by Example.

5. If I deliuer 340. li. ster. here in London after 6. s. 4. d. sterling the crowne, to receaue at Roan, or at Parris 50. sou. Tournois for euery crowne, I would knowe how many Liures Tournois I shall receaue

there for my 340. li. ster. Answere: say if 6. s. ⅓ doe giue mee 2. li. ½. Tour. what will 6800/1 shil. giue (which is the 340. li. reduced into shillings) multi∣plye and diuide, and you shall fynde 2684. Liures 4/19 which is worth 4. s. 4/19 Tournois, and so much shall I re∣ceaue in Roan or Parris for my 340. li. sterling.

6. If I deliuer in Parris or Roan, or elsewhere in Fraunce 1250. Liures Tournois, at 50. sou. Tournois the crowne to receaue for euerye suche crowne, 6. s. 3. d. sterling at London. I demaunde how much sterling mo∣ney I shal receaue at London for my 1250. pound Tournois. Answere: say, if two pounde, ½ doe giue mee 6. shil. ¼, what will 1250/1 giue? Multiplye & diuide, and you shall fynde 3125. shil. sterling, which maketh 156. pounde, 5. shillings sterling. And so manye poundes shall I receaue at London, for the sayde 1250. pounde Tour∣nois,

after 6. shillings three pence for euery crowne.

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

THe Rule of Alligation is so named, for that it tea∣cheth to alligate or bynde togither diuers percelles of sundrie prices, and to knowe how much you must take of euery percell, according to the numbers of the Question.

¶Example.

1. A Goldsmith hath three sortes of Golde. The fyrste is worth thirtye Crownes the pounde waight: The seconde is worth 36. Crownes. And the thyrde is worth 45. crownes, and of these three sortes hee wyll make a Scepter of syxe pound waight, which shall be worth fourtye Crownes the pounde. I demaunde how much hee must take of euery sorte?

Aunswere: fyrst you must set downe the numbers wherof you shall make y^e Alligation (which are 30.36.38. & 45. orderly the one vnder the other, as yf you should make of them an additiō: and the common number wherunto you will reduce them, shall you set on the left hand, which common nū∣ber in this exāple is 40. Then marke what summes bee lesser than that commō number, and which be grea∣ter, and with a draught of your pen, euermore lynke two numbers togi∣ther, so that the one be lesser thā that commō number, and the other grea∣ter than he. For two greater nor two smaller numbers may not be lynked togither, for they wil either be lesser, or else greater than the common nū∣ber: but one greater number, and one smaller may bee so mixed, y^t they wyl make the common number. And two greater or twoo smaller nūbers, can neuer make the common num∣ber in due order, as hereafter shall

appeare.

After that you haue thus lynked them, then marke howe much eche of the lesser numbers is smaller than the common number, and that diffe∣rence shal you set against the greater numbers, which be linked with those smaller, eche of them with his match still on the right hande. And likewise you must set the excesse of the greater numbers against the lesser which bee combined with them. Then shal you adde all those differences into one summe, which shalbe the fyrst num∣ber in the rule of three, and the second number shalbe y^e whole massye peece that you wyll haue of all the perticu∣lers, the thirde summe shall be eche difference by it selfe, and by thē shall you finde out the fourth number, de∣claring the iust portion of euery par∣ticuler in that mixture, as nowe by the former exaumple, I wyll make it playne.

The prices seuerall.	The diffe∣rences.
The common price or nūber.	40	30	5.	A
36	2.	B
42	4.	C
45	10	D
			21.

21.	.6.	.5.
21.	.6.	.2.
21.	.6.	.4.
21.	.6.	.10.

Here in this former example, you see that I haue set downe the seueral prices, whych bee 30.36.42.45. and haue linked togither 30. with 45. and 36. with 42. The common price 40. I haue set on y^e left syde, & the diffe∣rence of it from euery seuerall price, I haue sette on the righte hande, a∣gainst that summe with the which it is lynked. So the difference of 30. frō

fourtie is ten, which I set against 45. that hee is lynked with all, and the difference of 45. about 40 is 5. which I haue set againste. 30. So likewyse, the difference of 42. aboue 40. is 2. that I haue set against 36. And the differēce betweene 36. and 40. (which is 4.) I haue set againste 42. Then I adde all those differences togither & they make 21. which I make the fyrst number in the Rule of thre, and syxe the seconde number, which is the waight of the Scepter of Golde, and the thyrde number shall bee euerye particulers differēce. Then I worke by the Rule of three: saying if twenty and one (which is the differences ad∣ded togither) do giue mee syx pound, which ts the waight of the Scepter, what shall fyue giue, which is the fyrste difference?

Multiplye and diuide, and you shall fynde one pounde 3/7: so much muste I haue of the fyrste price.

Then doe likewyse with the rest and you shall finde 4/7 of the seconde price, one pound, 1/7 of the thirde price, and 2. li. 6/7 of the fourth, the which foure summes being added togither, doe make 6. li. which is the totall that I would haue. And now to proue if the pryces doe agree, you shall doe thus: Fyrst multiply this totall summe 6. by the common price 40. and it wyll make 240. crownes, which you shall keepe by it selfe. And afterward mul∣tiply euery seueral summe of waight by the price belonging to the same waight, and if that summe doe agre with the fyrst that you kept by it self, then is your work well done, as here one pound, 3/7 is the waight of y^e sorte of golde which is of 30. crownes price. Then multiplye 30. by 1. li. 3/7, & it ma∣keth 42. crownes 6/7, which you shall set downe. Then multiplye 4/7 (which is the wayght of the seconde sorte of gold) by 36. which is y^e price of y^e same & thereof commeth 20. crownes 4/7: so

againe 1. li. 1/7 multiplied by 42. doth make 48. crownes. And last of al 2. li. 6/7 multiplied by 45. maketh 128. crow∣nes 4/7. All these added togither doth make 240. crownes, agreable to the former summe of 40. multiplied by 6 And thus I maye affirme that thys worke is well done.

2. A Tauerner hath foure sortes of wyne, of foure seuerall prices, the fyrst of eight pence the Gallonde, the second of ten pence the gallonde, the thirde of 15. pence and the fourthe of 18. pēce. And he wil mingle one pun∣chen with all these sortes, so that the Gallonde shall bee worth but twelue pence. I demaunde how many Gal∣londes he muste take of euery sorte? Aunswere: First suppose the punchen to holde some certaine measure, as to conteine 84. gallonds and then the forme wyll be after thys sort, as you see hereafter folowing.

12	8	3
10	6
15	4
18	2
		15

If 15. doe giue 18.

What will. 3.	They make	16 ⅘ of the first.
What will. 6.	33 ⅗ of the secōde
What will. 4.	22 ⅖ of the third.
What will. 2.	11 ⅕ of the fourth
		84

¶The 14. chapter treateth of the Rule of falsehode, or false po∣sitions.

THe rule of falsehode is so na∣med, not for that it teacheth any deceit or falsehode, but y^t by fained numbers taken at al aduētures, it teacheth to finde out the true nūber y^t is demaunded. And this (of all the vulgare Rules which are in practise) is the most excellent: this rule hath two partes y^e one is of one false position alone: y^e other is of two positiōs as hereafter shal appere

Those questions which are done by false positions, haue theyr opera∣tions, in a maner like vnto that of the rule of three, but only that in the rule of three, we haue thre numbers kno∣wen, and here in thys rule wee haue but one (I meane that commeth in operation) vnto the likenes whereof wee muste diuise two other, the one multiplying, and the other diuiding, as by example.

1. I haue deliuered to a banker a certeine summe of poundes in mo∣ney, to haue of hym by the yeare 6. li. vpon the 100. li. And at the ende of 10. yeares, he paide mee 500. li. for al both principal and gaine. I demaund how much was the principall summe that I deliuered at the fyrst. Here you see that there are diuers termes: but the chiefe to worke with all is 500. pound which commeth of the other num∣bers, that is to saye, of 10. and 100. for of them is compounde the tenour

of the question, the practise whereof is thus.

Let vs faine a number at pleasure, and wyth the same let vs make oure discourse, euen as though it were the principall summe that wee seeke for. As by Example. Suppose that I de∣liuered him at y^e first 200. li. the which were worth to me in ten yeres. 120. li. after the rate of 6. li. vpō the hundred poūde. Thē 120. li. added with 200. li. Do make but 320. li. and I must haue 500. pound. Thus you see that I haue three termes for the rule of three: the one which shall conteine the Questiō the other two, which I haue formed artificially, which are 200. and 320: in such sort, that 320. ought to haue such proportion to 200, as 500. hath vnto y^e number that I seeke: that is to saye, vnto the true principall summe, then must I haue recourse vnto the rule of thre, after this sorte, saying.

If 320. li. become of 200. li. of howe much shall come 500. li. Multiply 500. by 200. and thei are 100000. the which you shall diuide by 320. li. and thereof commeth 312. li. ½ which is y^e summe that I deliuer at the first. 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 bee opened, the water wyll auoide cleane in one houre, at the seconde it wyll auoide in twoo houres, and at the thyrde it will require three houres. Nowe I demaūde in what space wil it auoide, all the cocks being set open. Suppose that it will auoide in halfe an houre, that is to say, in 30. minutes. Then must there auoide at the fyrste cocke the ½, which is thirtye 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 whych

summes being added togither dooe make 55. pipes, but it shoulde bee 60. pipes. Therefore I say by the rule of three, if 55. pipes doe voide in 30. mi∣nutes: in how many minuts wil 60. pipes voide? Multiplie and diuide, & you shall finde 32. minutes 40/55. And in that space will the water auoide if all the cockes be set open.

¶Of the Rule of twoo false po∣sitions.

* 1.13THe sūme of this Rule of two false positions is thus, when any questiō is proponed ap∣pertaining to this rule. First imagine any nūber at your pleasure, which you shal name the first positiō, and with the same shall you work in stede of the true number, as y^e questiō doth import, and if you see y^t you haue missed. Then is the last number of y^e work either to great or to little, that shal you note for to be y^e fyrst errour,

in the which you haue missed with y^e signe of more, or lesse, which signes shalbe noted with these figures, 4:—. This figure 4: betokeneth more, & this plaine line — signifieth lesse, that is to say the one signifieth to much, & the other to little: thē begin againe, & take another number, which shall be the second position, and worke by the questiō as before, if you haue missed againe, note the excesse or want, for that is the seconde errour. Thē shall you multiplie the first position by the second errour crossewise, and againe the seconde positiō by the first errour (& this must alwayes be obserued) & keepe the twoo products: then if the signes be both like, y^t is to say, either both to much, or both to little, abate the lesser product frō the greater, and likewise, you shall substract the lesser errour frō y^e greater, & by y^e remaine of those errours, you shall diuide the residue of y^e products: the quotiēt shal be the true nūber that you seeke for.

But if the twoo signes be vnlyke, y^t is to saye the one to much and the o∣ther to lyttle, thē shall you adde those productes togither so shall you also adde both the errours togither, and by y^e summe of those errours, diuide the totall sūme of both the products: the quotient shall bee lykewise the true number that the questiō seketh, and thys is the whole rule, as by exaumple.

3. A mā lying at the point of death, sayde that hee had in a certaine Cof∣fer a hundred Duckets, the whych he bequeathed to three of his frendes by hym named, after this sort. The first must haue a certeine portion, the se∣conde must haue twise so many as y^e fyrst abating eyght Duckets: and y^e thyrde must haue thre times so many as the fyrst, lesse by 15. Duckets. Now I demaunde howe many euerye of them must haue. Aunswere: Fyrst I doe imagine that the fyrst mā had 30.

Duckets, then by the order of y^e que∣stion the seconde should haue 52: & the third 75. These thre sūmes being ad∣ded togither do make 157: & I should haue but 100: so that this fyrst errour is to much by 57. then I note a parte the fyrst position 30. wyth his errour 57. to much after this sort 30.457. Therfore I prosecute my worke and I suppose that the fyrste had 24. then by the order of the questiō, the second should haue 40. and y^e thyrd 57: these three summes being added togither, do make 121. & I must haue but 100. so the seconde errour is to much by 21. Therfore I note 24.421. vnder the 30.457. as you may se in the margēt of the next syde following.

Then I multiplye crossewayes, 30. (whych is y^e fyrst position) by twelue whych is the seconde errour, and thereof commeth 630. lykewyse I multiplye twenty & 4. (whych is the seconde position) by 57. which is the fyrst errour, and I fynde 1368: then

bicause the signes of the errours are both lyke: that is to saye to much, I must therfore sub∣stract 630. from 1368. & there will remain 738 which is the diuidende: again I must sub∣stract the lesser er∣rour from y^e grea∣ter, that is to wit, 21. out of 57. and there wil remaine 36. which shall bee my diuisor. Thys done I diuide 738. by 36. and the quotient wyll be 20. ½. The which 20. ½ is the iust number of the Duckets that the first mā had for his parte, so consequently the seconde man had 33. Duckets, and the thyrde 46. ½, as by proofe in the margent, may appere. 〈 math 〉〈 math 〉

	11.
	738.
20 ½	366. (20. ½
33	3
46. ½
100.

The like number will also appere, in case the errours were both to litle, as in making the twoo positions by 18. and 20. where you shal fynde the two errours both to lyttle, the fyrst wyll bee to lyttle by 15. and the se∣conde to little by 3. as by perusing this worke in the margent you shal well perceiue. 〈 math 〉〈 math 〉

Againe if one of the errours were to muche, and the other to little, yet shall I haue the true number, as be∣fore: As if the twoo positions were 24. and 20. I shall fynde that the first errour will be 21. to muche, and the seconde wyll be three to little: Ther∣fore I multiply 24. by 3. crossewaies, therof commeth 72.

Lykewise I multi∣plye twentye by 21. the product will be 420: These twoo summes 72. and 420, I adde both togither bycause y^e signes of the er∣rours bee vnlyke, & they make 492. the which shall bee my diuidende, and againe, I adde the the lesser errour 3. wyth the greater errour 21. and they make 24. for my diuisor, thē diuiding 492 by 24. the quotient wilbe 20 ½: as in the margent doth plainly appeare. 〈 math 〉〈 math 〉

And now bicause you shall not for∣get thys part of the rule, learne thys briefe remembraunce folowing.

The signes both like substractiō do require. And vnlike signes addition will desire.

The meaning whereof is thus yf both the errours haue lyke sygnes, thē must the diuidende and the diui∣sor be made by substraction, as is taught before, and if those sygnes bee vnlike, then muste I by addition ga∣ther the diuidende, and the diuisor, as I haue done in this laste example.

4. A man hath twoo siluer cups of vnegall waight, hauinge to them bothe, but one couer, the weyght whereof is 5. ounces, if the couer bee put to the lesser cuppe, it wyll bee in double proportion vnto the waight of the greater, and the couer being put to the greater cuppe, wyll bee in tri∣ple proportion, vnto the waight of the lesser. I demaunde what was the waight of euerye cuppe?

Suppose that the lesser cuppe dyd wayghe seauen ounces, then wyth the couer it must waigh twelue, and thys waight should be in double pro∣portion vnto the greater, therefore

the greater muste waygh sixe ounces adde vnto it 5. oun∣ces for y^e couer, all will be 11. ounces, but it should be 21. for to haue it in tri∣ple proportion, vn∣to 7. which repre∣senteth the waight of the lesser cuppe: So that this fyrste errour is to little by 10. which you shal note after 7. in this sorte 7.— 10. 〈 math 〉〈 math 〉

After you shall suppose some other number, as 9. & make the lyke worke as before, so shal you find 15. to little, for y^e seconde errour, which you shall put behinde 9. and then worke with the reste as aboue is sayde, and you shall fynde that the lesser cup wayed three ounces, and consequentlye the greater foure ounces.

5. One man demaunded of ano∣ther

in a morning what a clocke it was, the other made hym this aun∣swere, if you doe adde (saith hee) the ¼ of the houres which be paste synce midnight, with the ⅔ of the houres which are to come vntyll noone, you shall haue the iuste houre, that is to say, you shall knowe what a clocke it was: Suppose y^t it was 4. a clocke in y^e morning, so shoulde there remaine 8. vntill none: then I take the ¼ of 4. which is 1. and the ⅔ of eight which is 5. ⅓, and I adde them togither, so I finde 6. ⅓ and supposed but 4. there∣fore this fyrst error is to much by 2. ⅓, which I note after my position thus, 4.42. ⅓: then againe I suppose ano∣thee nūber, y^t is to saye 9. so shoulde remaine but 3. houres vntill none, I take the ¼ of 9. and the ⅔ of 3. which is 2. ¼ & 2. these I adde togither and they make 4. ¼: but I supposed y^t it was 9. therefore the seconde errour is 4. ¾ to little which I note behinde my posi∣tion thus 9.44. ¾.

And then I multiplie crossewyse, as before is taught, and bicause the sygnes of the er∣rours are vnlike, that is to saye, the one to much, and the other to lyttle, therefore in thys worke I muste adde the productes, and they wyll bee 40. Lykewyse I adde the errours, and they be 7.1/12. Then I diuide 40. by 7.1/12, and thereof commeth 5. houres 11/17, and that houre it was in the mor∣nyng. 〈 math 〉〈 math 〉

¶The 15. Chapter treateth of sportes, and pastime, done by number.

IF you would know the num∣ber that anye man doth thinke or imagine in hys minde, as though you coulde deuine.

Bydde him triple the same number, then of the product let him take the, ½

if the number bee euen, or else the greater halfe, if y^e same be odde, then bid hym triple againe the saide ½: af∣ter say to him that put away, if he cā 36.27.18. or 9. from the last number being tripled: y^t is to saye, cause hym subtellye to put awaye 9. as many times as is possible & keepe the nūber secretly: & when he can no more take away 9. thē to know if y^t yet there re∣maine any nūber, bid him abate 3.2. or one, if he can: this done, see howe many times 9. you haue caused hym to abate, for the which keepe you in minde so many times 2. & if that you know y^t he had any thing remaining besyde the nines, the same shall also note vnto you 1.

Suppose that he thought 6. which being tripled is 18. wherof the ½ is 9. the triple of y^t is 27: nowe cause him to abate 18, or 9. or 27. and againe 9: but then he will say vnto you that he cannot, bid him then abate 3. or 2. or 1. he will say also that he can not: wher∣fore

considering that you haue made him to abate three times 9. iustly, you shal tell him that he thought 6. for 3. times 2. maketh 6. If he had thought 5. the triple thereof is 15. whereof the greater ½ is 8. the triple of y^t maketh 24. which cōteineth two times 9. thei are worth 4. & the remaine signifieth 1. the which added togither make 5. which is y^e number that he thought.

2. If in any company, one of them hath a ringe vpō his finger, and you would know by maner of deuining, who hath the same & vpon what fin∣ger & what ioint: cause the persons to sit down in order, & kepe likewise an order of their fingers: then seperate your self from them in some certeine place, and say vnto one of the lookers on, that he double the number (mar∣king the order) of him y^t hath y^e ring: and vnto the double bid hym adde 5. and then cause him to multiplye this addition by 5. & vnto the product byd him adde the number of the finger of

the person which hath the ring: be it y^t the same laste sum did amoūt to 89. then afterwarde saye to him y^t he put after the same last nūber toward hys right hande a figure signifiyng vpon which of the ioints he hath the ring. As if it bee vpon the thirde ioint, let him put 3. after 89. and it wil be 893: this done, you shall aske him what number hee kepeth, from the which you shall abate 250. & you shall haue three fygures remaining at the least. The fyrste towarde your lefte hande shall signifye the number of the per∣son which hath the ring. The second or middle fygure shall represent the number of the finger. And the last fy∣gure towarde your right hande, shal betoken the number of the ioint. As if the number which hee did keepe were 893. from that you shall abate 250. and there wyll remaine 643: Which do note vnto you, that y^e syxt person hath the ring vpon the fourth finger, and vpon his thirde iointe.

But note y^t when you haue made your substraction, if there do remain a cipher in the place of tennes, y^t is to say in y^e seconde place, you must then abate from that figure which is in y^e place of hundreds, y^t is to say from the figure which is next your left hand, & y^t shalbe worth ten tenths, signifiyng the tenth finger: as if there should re∣maine 703. you must say that the syxt person (vpon his tenth finger & vpon hys thirde ioint) hath the ring.

3. And after the same maner, if a mā do cast three dice, you may know the poinctes of euery one of thē, for if you do cause him to double y^e poincts of one die, & vnto y^e double to adde 5. & the same sum to multiply by 5. & vnto y^e product adde the poincts of one of y^e other dice, and behinde the number toward y^e right hand, to put y^e figure which signifieth y^e poincts of the last die, & thē shal you aske him what nū∣ber he kepeth frō y^e which abate 250. & there wil remaine 3. figures, which do

note vnto you y^e poincts of euery dye.

4. Likewise if 3. of your compani∣ons, to say, Peter, Iames, and Iohn that woulde (in your absence) giue themself euery one a contrary name: as for example: Peter would be cal∣led a king, Iames a duke, and Iohn a coūtie: And I would deuine which of them is called a king, which the duke, and which the countie. Take 24. stones, or other peeces whatsoe∣uer, & giue vnto Peter 1. vnto Iames 2. and vnto Iohn 3. or otherwise. But marke well vnto which of them you haue giuen 1, vnto which 2, and vnto whom 3. Then leauing the 18. stones (before thē) that are remaining, you shall absent your selfe from theire syght, or else tourne your face from them, saying thus vnto them, who∣soeuer nameth hymselfe a king: for euery stone that I gaue him let hym take one of the residue, and hee that nameth himselfe a duke for euerye stone that I gaue him let him take 2

of them that remain, and he that cal∣leth himself a countie, for euery stone that I gaue him let him take 4: this being done approche nere them, and marke how many stones are remai∣ning: and know this, that there can not remaine any other number, but one of these sixe, 1, 2, 3, 5, 6, 7, for the which sixe numbers we haue chosen to euery of them a seuerall name, which are these: Angeli, Beati, Quali∣ter, Messias, Israel, Pietas: eche of them containing three Vowelles, a, e, i, which doe shew the names by order: That is to say, A,

1	2	1	2	3	3
2	1	3	3	1	2
3	3	2	1	2	1
a	e	a	e	i	i
e	a	i	i	a	e
i	i	e	a	e	a
1	2	3	5	6	7
A	B	Q	M	I	P

sheweth which is the kinge, E, tel∣leth which is the duke and, I, she∣weth which is y^e county: in follo∣winge the order how, & to whom you haue giuen one stone, to which 2, and to which 3.

Then if there doe remaine but one stone, the first name Angeli, (by these thre vowels a, e, i,) sheweth that Pe∣ter is the king, Iames the duke, and Iohn the countie. And if there doe remaine 2 stones, the seconde name Beati, shal shew you by these three vowels, e, a, i, that Peter is y^e Duke, Iames the King, and Iohn the countie. And so of the other as by this Table doth plainely ap∣pere.

FINIS.

Notes

• * 1.1

  A place.
• * 1.2

  Ternaries.
• * 1.3

  Three kin∣des of Nū¦ber.
• * 1.4

  Diget.
• * 1.5

  Article.
• * 1.6

  Mixt or cō∣pounde.
• * 1.7

  A note.
• * 1.8

  A note.
• * 1.9

  Progression arithmetical.
• * 1.10

  Progession, geometricall.
• * 1.11

  Regul.
• * 1.12

  A Rule.
• * 1.13

  A Rule.