The grounde of artes teaching the perfect vvorke and practise of arithmetike, both in whole nu[m]bers and fractions, after a more easie ane exact sort, than hitherto hath bene set forth. Made by M. Robert Recorde, D. in Physick, and afterwards augmented by M. Iohn Dee. And now lately diligently corrected, [and] beautified with some new rules and necessarie additions: and further endowed with a thirde part, of rules of practize, abridged into a briefer methode than hitherto hath bene published: with diverse such necessary rules, as are incident to the trade of merchandize. Whereunto are also added diuers tables [and] instructions ... By Iohn Mellis of Southwark, scholemaster.

The second Dialogue. The accompting by Counters.

Mayster.

NOw that you haue learned the commō kindes of Arithme∣tike with y^e pen, you shall sée the same art in Counters: whiche feate doth not onely serue for them that cannot write and reade, but also for them that can doe both, but haue not at some times their pen or tables readie with them.

This sort is in two formes commonlye: The one by lines, and the other withoute lines. In that that hath lines, the lines doe stande for the order of places: and in that that hath no lines, there must be set in their steade so many counters as shall néede, for ••che line one, and they shall supplie y^e steade

of the lines.

Sc.

By examples I should better perceiue your meaning.

Mai.

For example of 〈 math 〉〈 math 〉 the Lynes, loe héere you see sixe lines, which stande for sixe places, so that the nethermoste standeth for the firste place, and the next aboue it for the seconde, and so vpwarde, til you come to the highest, which is the sixt line, and standeth for the sixt place.

Now what is the value of euery place or line you may perceiue by the figures which I haue set on them, whiche is according as you learned before in Numeration of Fi-figures by the pen: for, the firste place is the place of vnits or ones, and euerye Counter set in that line betokeneth but one: and the second line is the place of 10, for euery coū∣ter there standeth for 10. The thirde lyne the place of hundreds, y^e fourth of thousāds, and so forth.

Sc.

Sir, I doe perceiue that the same or∣der is here of lines, as was in the other fi∣gures by places, so that you shall not néede

longer to stande about Numeration, except there be any other difference.

M.

If you doe vnderstande it, then 〈 math 〉〈 math 〉 how will you set 1543?

Sc.

Thus as I suppose.

Ma.

You haue set the places truly but your figures be not méete for

[illustration]

this vse: for the méetest figure in this behalf, is the figure of a Counter round, as you sée here, where I haue expressed that same summe.

Scholer.

So that you haue not one figure for 2 nor 3, nor 4 and so forth, but as ma∣ny digits as you haue, so many Counters you set in the lowest line: and for euery 101 you sette one in the seconde line: and so of o∣ther. But I knowe not by what reason you set that one counter for 500 betwéene two lines.

Maist.

You shall remember this, that whensoeuer you néede to sette downe 5, 50, or 500 or 5000, or so forth any number whose Numerator is 5, you shall sette one counter for it in the next space aboue y^e line that it hath his denomination of: as in this

example of that 500, because the numera∣tor is 5, it must be set in a voyde space: and bicause y^e denominator is hundred, I know that his place is the voyde space next aboue hundreds, that is to say, aboue the thirde line.

And farther you shall marke, that in all working by this sorte, if you shall set down any summe between 4 and

[illustration]

10, for the first part of that number you shal set downe 5, and then so many Coun∣ters more, as there rest nū∣bers aboue 5. And this is true both of digits and ar∣ticles. And for example I will set downe this summe 287965, whiche summe if you marke well you néede none other examples for to learne the Nu∣ration of this forme,

But this shall you marke, that as you didde in the other kindes of Arithmetike, sette a pricke in the places of thousandes, in this worke you shal set a Starre, as you sée before.

Scholer.

Then I perceiue Numeration: but I pray you how shall I doe in this art to adde two summes or more togither.

ADDITION.

Maister.

THe easiest waye in this art, is to ad but two sums at once togither: howbeit, you may add more, as I wil tel you a∣non. Therefore whē you wil add 2 sums you shall first set down one of thē, it forceth not which, and then by it draw a line crosse the other lines. And afterward set down the other sum, so that that line may be between them:* 1.1 as if you woulde

[illustration]

add 2659 to 8342, you must set your sums as you sée here.

And then if you list, you maye adde the one

to the other in the same place: or else you may ad them both togither in a new place: which way, because it is most playnest, I will shew you first.

Therefore will I beginne at the vnits, which in the first sume is but 2, and in the second summe 9, that maketh 11. Those do I take vp, and for them I sette 11 in the new roome, thus.

[illustration]

Then doe I take vp all the Articles vn∣der a hundred, which in the first summe are 40, and in the second summe 50, that ma∣keth 90: or you may say better, that in the first summe there are 4 articles of 10, and in the seconde summe 5, whiche maketh 9, but then take héede that you set thē in their

right lines as you sée here.

[illustration]

Where I haue taken away 40 from the firste summe, and 50 from the seconde, and in their stéede I haue sette 90 in the thirde roome, which I haue set plainelye that you might well perceiue it: how be it, seeing that 90 with the 10 that was in the thirde roome already, doth make

[illustration]

100, I might better for those 6 Counters set 1 in the thirde line, thus.

For it is all in one sum as you maye sée, but it is best neuer to sette fyue counters in any line, for that maye be done with one Counter in a higher place.

Scholer.

I iudge that good reason, for manye are vnnéedefull where one will serue.

Maister.

Well, then will I adde forth of hundreds: I finde 3 in the first summe, and 6 in the second which maketh 900, them do I take vp, and set in the third roome, where is one hundred alreadie, to whiche I putte 900 and it will be 1000, therfore I set one counter in the fourth line for them all, as you sée here.

[illustration]

Then adde I the thousandes together, which in the first summe are 8000, and in the second 2000, that maketh 10000: them doe I take vp from those two places, and for them I set one counter in the fifte line, and then appeareth as you sée to be 11001,

[illustration]

for so manye doth a∣mount of the Addition of 8342 to 2659.

Scholer.

Syr,* 1.2 this I doe perceiue: but howe shall I sette one sūme to an other, not chaunging them to a third place?

Mai.

Marke well how I doe it: I will adde togither 65436 and 3245, which firste I set down thus.

[illustration]

Then do I begin with the smallest, which in the first sūme is 5, that do I take vp, and would put to the other 5 in the second sum,

sauing that two Counters cannot be set in a voyd place of 5 but for them both I must set 1 in the second lyne, which is the place of 10: therefore I take vp the fiue of the first summe, and the 5 of the second, and for them I set 1 in the secounde line, as you see here.

[illustration]

Then doe I likewise take vp the 4 coun∣ters of the first sum and second line (which make 40) and ad them to the 4 counters of the same line, in the seconde summe, and it maketh 80, but as I sayde, I maye not conuenientlye set aboue 4 counters in one line, therefore to those 4 that I tooke vp in the first summe, I take one also of the se∣cond

summe, and then haue I taken vp 50, for which 5 Counters I sette down one in the space ouer the second line, as here doth appeare.

[illustration]

And then is there 80, as well with those 4 counters, as if you had set downe the other 4 also.

Now do I take the 200 in the first sum, and adde them to the 40 in the second sum, and it maketh 600, therefore I take vp the 2 counters in the first summe, and 3 of thē in the second summe, and for them 5, I set 1 in the space aboue, thus.

[illustration]

Then I take the 3000 in the first sūme, vnto which there are none in the seconde summe agréeing, therefore I doe onelye remoue those thrée Counters from the first sum into the seconde, as here doth ap∣peare.

[illustration]

And so you sée the whole summe that amounteth of the Addition of 65416 with 3245, to bée 68681.

And if you haue marked these two exam∣ples well, you neede no further instruction in Addition of 2 onely summes: but if you haue more than two summes to adde, you may ad them thus.

Firste adde two of them, and then adde the thirde and the fourth, or more if there be so many: as if I woulde adde 2679 with 4286 and 1391. First I adde the two first summes thus.

[illustration]

And then I adde the third therto thus.

[illustration]

And so of more, if you haue them.

Scho.

Nowe I thinke beste that you passe

forthe to Subtraction, excepte there be anye wayes to examine this manner of Addition, then I thinke that were good to be knowen next.

Maister.

* 1.3There is the same proofe here that is in the other Addition by the penne, I meane Subtraction, for that onelie is a sure waye: but considering that Subtraction muste be first knowen, I will first teach you the arte of Subtraction, and that by this ex∣ample.

SVBTRACTION.

I Woulde subtracte 2892 out of 8746. These summes muste I sette downe as I did in Addition: but here it is beste to sette the lesser number first thus.

[illustration]

Then shal I beginne to subtract the grea∣test numbers first (contrarie to the vse of the pen) that is the thousandes in this example: therefore I finde amongst the thousands 2, for which I withdraw so manye from the se∣cond summe (where are 8) and so remayneth there 6, as this example sheweth.

[illustration]

Then do I likewayes with the hundreds, of whiche in the firste summe I finde 8, and in the seconde summe but 7, out of whyche I can not take 8, therefore this must I doe: I muste looke howe muche my summe diffe∣reth from 10, whiche I finde here to be 2, then muste I bate for my summe of 800, one thousande, and set downe the excesse of hun∣dreds, that is to saye, 2, for so muche 1000 is more than I shoulde take vp. Therefore from the firste summe I take that 800, and from the seconde summe (where are 6000) I take vppe one thousande, and leaue 5000, but then set I downe the 200, vnto the 700 that are there alreadie, and make them 900, thus.

[illustration]

Then come I to the articles of tennes, where in the firste summe I finde 90, and in

the seconde summe but onely 40. Now consi∣dering that 90 can be bated from 40, I looke how muche that 90 doth differ from the next summe aboue it, that is 100, or else (which is all to one effecte) I looke howe much 9 doeth differ from 10, and I finde it to be 1, then in the steade of that 90, I doe take from the seconde summe 100: but considering that is 10 too muche, I set downe 1 in the nexte line beneth for it, as you sée

[illustration]

here.

Sauing that here I haue set one counter in the space, in steade of 5, in the next line.

And thus haue I subtract al, saue two whiche I muste bate from the 6 in the se∣cond

[illustration]

summe and there will remaine 4, thus.

So that if I subtract 2892 from 8746, the remainer wil be 5854

And that this is tru∣lie wrought you may proue by addition: for if you adde to this remainer the same summe

that you did subtracte, then will the former summe 8746, amount againe.

Scho.

That wil I proue: and first I set the sum that was Subtracted, which was 2892, and then the remainer 5854, thus.

[illustration]

Then doe I adde the first 2 to 4, which ma∣keth 6: so take I vp 5 of those counters, and in their stead I set 1 in the space, and 1 in the lowest line, as here appeareth.

[illustration]

Then doe I adde the 90 next aboue to the 50, and it maketh 140, therefore I take vppe those 6 counters, and for them I set 1, to the hundreds in the thirde line, and foure in the second line thus.

[illustration]

Then do I come to the hundreds, of which I finde 8 in the first sum, & 9 in the seconde, y^t maketh 1700: therefore I take vp those 9 counters, & in their steade, I set 1 in y^e fourth line, and 1 in the space next beneath, and 2 in the third line as you sée here.

[illustration]

Then is there lefte in the firste summe but onelie 2000, whiche I shall take vppe from thence, and set in the same line in the

[illustration]

seconde summe, to the one that is there al∣readie: and then wyll the whole summe ap∣peare as you maye well sée, to be 8746, whiche was the firste grosse summe, and therefore I do perceiue that I had wel subtrac∣ted before.

And thus you may sée, howe Subtraction may be tried by Addition.

Scho.

I perceiue the same order here wyth Counters, that I learned before in figures.

Ma.

Then let me sée how you can trie Ad∣dition by Subtraction.

Scholer.

Firste I will sette forth thys ex∣ample of Additiō, where I haue added 2189, to 4988. And the whole summe appeareth to be 7177.

[illustration]

Nowe to trie whether that summe be wel added or no, I wil subtract one of the first two summes from the thirde, and if I haue well done, y^e remainer wil be like that other sum, as for example. I wil subtracte the first sum from the thirde, which I set thus in their or∣der.

[illustration]

Then doe I subtract 2000 of the firste sum from the second summe, and then remaineth there 5000, thus.

[illustration]

Then in the third line I subtracte the 100 of the firste sum from the seconde sum where is onely 100 also: and thē in the third line resteth nothing, as you maye sée in the example following.

[illustration]

Then the in seconde line with his space o∣uer him, I finde 80, whiche I shoulde sub∣tract from the other sū then séeing ther are but onelie 70, I must take it out of some higher sum, which is here only 5000: therefore I take vp 5000: and séeing that is too much by 4920, I set down so many in the second roome, which with the 70 being there alreadie, do make 4990, and then the summes do stand thus.

[illustration]

Yet remaineth there in the firste sum, 9 to be abated from the second sum, wherein that place of vnits doth appeare only 7: then must I bate a higher summe, that is to saye 10, but séeing that 10 is more than 9 (whiche I shoulde abate) by 1, therefore shall I take vp one counter from the

[illustration]

second, and set down the same in the firste or lowest line, as you see here.

And so haue I en∣ded this worke, and the summe appea∣reth to be the same whyche was the se∣conde summe of mine Addition, and therefore I perceiue I

haue wel done.

May

To stande longer about this, it is but folly,* 1.4 except that this you may also vnderstād, that many do begin to subtract with counters not at the highest summe as I haue taughte you, but at the neathermost, as they do vse to adds: and when the summe to be abated in a∣nye line appeareth greater than the other, thē do they borrow one of the next higher roome, as for example.

[illustration]

If I shoulde abate 1846 from 2378, they set the summes thus.

First they take 6, whiche is the lower line, and his space, from 8 in the same roomes in the seconde summe, and yet there remay∣neth 2 counters in the lowest line. Then in the seconde line must 4 be subtracted from 7, and so remaineth there 3. Then 800 in the thirde line, and his space, from 300 of the se∣conde summe can not be, therefore doe they bate it frō a higher roome, that is from 1000: and because that 1000 is too muche by ••0••, therfore must I set downe 200 in the thirde line, after I haue taken vp 1000 from the

fourth line. Then is there yet 1000 in the fourth line of the first sum, whiche if I with∣drawe from the seconde sum, then doth al the figures stand in order, thus.

So that (as you

[illustration]

sée) it differeth not greatly whe∣ther you beginne subtraction at the higher lines, or at the lower.

Howe be it, as some men like y^e one way beste, so some like the other: therefore you nowe knowing bothe, maye vse whyche you liste.

MVLTIPLICA∣TION.

BVt nowe touchyng Multiplication: you shall sette your num∣bers in two roomes (as you did in those other kindes) but 2 so y^t the multiplier be set in the first roome, then shall you begin with the highest numbers of the seconde roome, and multiplie them firste, after this sort. Take y^e ouermost line in your first working,* 1.5 as if it were the lowest line, setting on it some moueable marke (as you list) and looke how many counters be in him, take them vp, and for them sette downe the whole multiplier so many times as you tooke vp counters: reckning (I say) that line for the Vnites. And when you haue done with the highest number, then come to the nexte lyne beneath, and do euen so with it, and so with the nexte, till you haue done all. And if there be anye number in a space, then for it shall

you take the multiplier 5 times: and then muste you recken that line for the Vnites, whiche is next beneath that space. Or else after a shorter waye, you shall take onelie halfe the multiplier, but thē shal you take the line next aboue that space for the line of Vnites. But in suche working, if by chaunce your multi∣rlyer be an odde number, so that you can not take the halfe of it iustlye, then muste you take the greater halfe, and set downe that, as if that it were the iuste halfe: and further you shall sette one Counter in the space beneath that line, which you recken for the line of V∣nits, or else onely remoue forwarde the same that is to be multiplied.

S.

If you set forth an exāple hereto, I think I shal perceiue you.

Maaster.

Take

[illustration]

this example: I woulde multi∣plie 1 5 4 2 by 2 6 5, therfore I set the numbers thus.

Then firste I beginne at the 1000 in the highest roome, as if it were the first place, and I take it vp, setting downe for it so often (that is once) the multiplyer, which is 365, thus as you sée here: where, for the one counter ta∣ken vp from the fourth line, I haue set down other 6, which make y^e sum of the multiplier, reckening that fourth line as if it were the first, which thing I haue marked by the hand set at the beginning of the same.

[illustration]

Scholer.

I perceiue this well, for in déede this sum that you haue set down is 265000: for so much doth amounte of 1000, multiplied by 365.

Maister.

Well, then to goe foorth, in the

next space I finde one counter, whiche I re∣moue forward, but take it not vp, but doe (as in such case I must) set down the greater half of my multiplier (séeing it is an od number) which is 182, and here I doe still lette that fourth place stand, as if it were the first: as in these examples you shall sée.

[illustration]

* 1.6 Where I haue set this multiplication with other, but for the ease of your vnderstāding, I haue set a little line betwéene them. Now should they both in one summe stand thus.

[illustration]

* 1.7

* 1.8

[illustration]

Howbeit, an other fourme to multiplye such coūters in space, is this: Firste to re∣moue y^e finger to the line next beneth that space, and thē to take vp that Counter, and to set down the mul∣tiplier fiue times: as here you see.

Which summes if you do adde togither into one summe, you shall perceiue that it will be the same that appeareth of the o∣ther working before, so that both sorts are to one intente: but as the other is shor∣ter, so this is play∣ner to reason for such as haue had smal ex∣ercise in this arte. Notwithstāding you

may adde them in your minde before you set them downe: as in this example you might haue sayde, 5 times 300 is 1500, and 5 times 60 is ••00 also 5 times 5 is 25, which all put togither, doe make 1825, whiche you may at one time set downe if you list.

But now to go forth, I must remoue the hande to the next counters whiche are in the second lyne, and there must I take vp those 4 counters setting downe for them my mul∣tiplier 4 times seuerally, or else I maye ga∣ther that whole sum in my mynde firste, and then set it downe: as to say, 4 times 300 is 1200: 4 times 60 are 240: and 4 times 5 make 20, that is in all 1460, that shall I set downe also, as here you sée.

[illustration]

Which if I ioyne in one sum with the for∣mer

numbers it will appeare thus,

[illustration]

Then to ende this Multiplication, I re∣moue the finger to the lowest line, where are only 2, them do I take vp, and in their stéede doe I set downe twice 365, that is 730, for which I set one in the space aboue the thirde line for 500, and 2 more in the thirde lyne with that one that is there alreadie, and the rest in their order, and so haue I ended the whole summe, thus.

[illustration]

Whereby you sée, that 1542 (whiche is the number of yeares sith Christe his incarnati∣on) being multiplyed by 365 (which is the number of dayes in one yeare) doth amounte vnto 562830,* 1.9 which declareth the nūber of dayes sith Christes incarnation vnto the end of 1542 yeares, (beside 385 dayes and 12 houres for leape yeares.)

Sc.

Now will I prooue by an other exam∣ple, as this: 40 labourers (after 6 d the day for eche man) haue wroughte 28 dayes: I would know what their wages doth amoūt vnto.

In this case must I worke doublye: first I must multiplie the number of the labou∣rers by the wages of a man for one daye, so will the charge of one day amount. Then secōdarily shal I multiplie the charge of one daye by the

[illustration]

whole number of dayes, and so wil y^e whole sum appear: first therfore I shal set the sūmes thus:

Where in the firste place is the Multiplier (that is 1 dayes wages for one man) & in the seconde space is

set the nūber of y^e warkmē to be multiplied.

Then saye I: 6 times 4 (reckoning that second line of the line of Vnits) maketh 24, for which summe I

[illustration]

should set 2 counters in the third line, and 4 in the second there∣fore doe I set 2 in the third line, and let the 4 stand still in the se∣cond line thus.

So appeareth the whole dayes wages to be 240 d y^t is 20 s.

Then doe I multi∣plye

[illustration]

agayne the same summe by the nūber of dayes, and firste I set the nūbers thus, Thē because ther are

[illustration]

counters in diuerse lines, I shall begin with the highest, and take them vp, setting for them the multi∣plier so many times as I tooke vp coun∣ters, that is twise, then wil y^e sum stād thus.

Then come I to the second line, and take vp those 4 Counters, setting for thē the mul∣tiplier foure times, so

[illustration]

wil the whole summe appeare thus.

So is the whole wa∣ges of 40 workemen for 28 days (after 6 d eche daye for a man) 6720 d that is 560 s or 28 pound.

Ma.

Now if you would prooue Multipli∣cation, the surest way is by Diuision: there∣fore will I ouerpasse it, till I haue taughte you the arte of Diuision, whiche, you shall worke thus.

DIVISION.

FIrst set downe the diuisor, for feare of forgetting, and thē set the nūber that shal be diuided, at the right side, so farre from the Diuisor, that the quotient may be set betwéene them: as for example.

If 225 shéepe cost 45 lb. what did euerye

shéepe cost? To know this, I shoulde diuide the whole summe that is 45 lb, by 225, but that cannot be: therefore must I firste reduce that 45 lb into a lesser denomination, as into shillinges, then I multiplie 45 by 20, and it is 900: that summe shall I diuide by the number of sheepe, which is 225, these two numbers therefore I set thus.

[illustration]

Then begin I at the highest lyne of the di∣uident, and séeke how often I maye haue the diuisour therein, and that maye I doe foure times: then saye I, foure times 2 are 8, whi∣the if I take from 9, there resteth but 1, thus.

[illustration]

And because I founde the diuisor 4 times in the diuident, I haue set as you sée, 4 in the middle roome, which is the place of the quotient: but now must I take the rest of y^e diuisour as often out of the remayner, there∣fore come I to the seconde line of the diuisor, saying: 2 foure times make 8, take 8 from 10, and there resteth 2, thus.

[illustration]

Then come I to the lowest number which is 5, and multiplie it 4 times, so is it 20, that take I from 20, & there remayneth nothing,* 1.10 so that I sée my quotient to be 4, which are in valewe shillings, for so was the diuident: and therby I know y^t if 225 Shéepe did cost 45 lb, euery shéepe cost 4 s.

Sch.

This can I doe,* 1.11 as you shall per∣ceiue by this exāple. If 100 soldiours do spēd euery moneth 68 lb, what spendeth ech man?

First because I cannot diuide the 68, by 160, therefore I will turne the lb into pen∣nies

by multiplicatiō, so shal there be 16320 d. Now must I diuide this summe by the number of souldiors, therefore I set them in order thus.

[illustration]

Then beginne I at the highest place of the diuidend, séeking my Diuisor there, which I finde once, therefore sette I 1 in the nether line.

Ma.

Not in the nether line of the whole summe, but in the nether lyne of that worke which is the third line.

Sc.

So standeth it with reason.

Ma.

Then thus doe they stand.

[illustration]

Then séeke I agayne the rest, how often I may finde my diuisour: and I sée that in the 300 I mighte finde 100 thrée tymes, but then the 60 will not be so often found in ••0, therefore I take 2 for my quotient: thē take I 100 twice frō 300, and there resteth 100, out of which with the 20 (that maketh 120) I may take 60 also twice, and then standeth the numbers thus.

[illustration]

Where I haue set the quotient 2 in the lowest line: So is euery Souldiors portion 102 d that is 8 s, 6 d.

Ma.

But yet because you shall iustlye perceiue the reason of Diuision, it shall bée good that you doe set your diuisor stil agaynst those numbers from which you doe take it, as by this example I will declare.

If the purchase of 20 acres of ground did cost 290 pound, what did one acre cost?* 1.12

First will I turne the poundes into pen∣nies,

so will there be 69600 pence. Then in setting down these numbers, I shal do thus. First set the diuidend on the right hand as it ought, and then the diuisor on the lefte hande agaynst those numbers from which I intend to take him firste as here you sée, where I haue set the diuisor two lines higher than is his owne place.

[illustration]

Scho.

This is like the order of Diuision by the pen.

Ma.

Truth you say, and now must I sette the quotient of this worke in the thirde line, for that is the lyne of vnits in respecte to the diuisor in this worke.

Then I séeke how often the diuisor maye be found in the diuident, and that I fynd 3 tymes, then set I 3 in the third lyne for the quotient and take awaye that 60000 from the diuidend, and farther I do set the diuisor

one line lower, as you see here.

[illustration]

And then séeke I howe often the diuisor will be taken from the number agaynste it, which will be 4 times and 1 remaining.

Scho.

But what if it chaunce that when the diuisor is so remoued, it cannot be ones taken out of the diuident against it?

Ma.

Then muste the diuisour be set in an other line lower.

Sc.

So was it in diuision by the pen, and therefore was there a cipher set in the quoti∣ent: but how shall that be noted here?

Ma.

Here néedeth no token, for the lines doe represent the places: onely looke that you set your quotient in that place whiche stan∣deth for vnits in respect of y^e diuisor: but now to returne to the example. I finde the diuisor 4 times in the diuident, and 1 remaining, for 4 times 2 make 8, which I take frō 9, & there

resteth 1, as this figure following sheweth; and in the middle space for the quotient I set 4 in the second line 〈◊〉〈◊〉 he is in this worke the place of vnits.

* 1.13

[illustration]

Then remoue I the diuisour to the next lo∣wer line, & seeke how often I may haue it in the diuident, which I may doe here 8 tymes iust, and nothing remain, as in this fourme.

[illustration]

Where you may sée, that the whole quoti∣ent is 348 d, that is 29 s, whereby I knowe that so much cost the purchase of one acre.

Sc.

Now resteth the proues of Multiplica∣tion, and also of Diuision.

Ma.

Their best proues are eche one by the other: for multiplication is prooued by Di∣uision, and Diuision by Multiplication, as in the worke by the pen you learned.

Sc.

If that be all, you shall not néede to re∣peate agayn that that was sufficiētly taught alreadie: and except you will teache me any other feate, here maye you make an ende of this art, I suppose.

Ma.

So will I doe as touching whole number: and as for brokē number, I will not trouble your wit with it, till you haue pra∣ctised this so well, that you be full perfect, so that you néede not to doubt in any point that I haue taught you, and then maye I boldlye instruct you in the arte of Fractions or Bro∣ken number: wherein I will also shew you the reasons of al that you haue now learned. But yet before I make an end, I will shew you the order of common casting, wherin are both pennies, shillings, and poundes, procée∣ding by no grounded reason, but onelye by a receyued forme, and that diuerslye of diuerse men: for the Marchantes vse one forme, and Auditours an other.

Marchantes vse.

BVt first for Mar∣chantes

[illustration]

fourme, marke this ex∣ample here, in whiche I haue expressed this sum 198 lb 19 s 11 d. So that you maye sée that the lowest line serueth for pen∣nies, the nexte aboue for shillings, the third for pounds, and the fourth for scores of poun∣des.

And further you may sée, that the space be∣twéene d and s may receiue but one counter (as al other spaces likewise do) and that one standeth in that place for 6 d.

Likewise betwéene the shillinges and the poundes, one counter standeth for 10 s.

And betwéene the poundes and 20 lb. one counter standeth for 10 lb.

But beside those you maye sée at the lefte side of shillinges, that one counter standeth alone, and betokeneth 5 s.

So agaynst the poundes, that one counter standeth for 5 lb. And against the 20 pounds, the one counter standeth for 5 score pounds, that is 100 pounde, so that euery side coūter

is 5 times so muche as one of them againste which he standeth.

Auditours Accompt.

NOwe for the accompte of Auditours, take this example.

[illustration]

Where I haue expressed the same summe 198 lb 19 s 11 d.* 1.14

But here you see the pence stand towardes the right hande, and the other increasing or∣derlie toward the left hande.

Againe you maye sée that Auditours wyll make 2 lines (yea and more) for pence, shil∣lings, and all other values, if their summes extende thereto. Also you see that they set one counter at the right ende of each rowe, which so set there, standeth for 5 of that roome: and on the lefte corner of the rowe it standeth for 10 of the same row.

But now if you would adde other subtract after any of both those sortes, if you marke the order of the other feate which I taughte you, you may easilie doe y^e same here without

muche teaching: for in Addition you muste firste sette down one summe, and to the same set the other orderlie, and in like maner if you haue manye: but in Subtraction you muste set downe first the greatest summe, and from it muste you abate the other, euerie denomi∣nation from his due place.

Sc.

I do not doubt but with a little practise I shal attain these both: but how shal I mul∣tiplie and diuide after these formes?

M.

You can not duelie doe anye of both by these sortes, therefore in such case you muste resort to your other artes.

S.

Sir, yet I sée not by these sortes howe to expresse hundreds, if they excéed one hundred, neither yet thousands.

Maister.

They that vse suche accomptes that it excéede 200 in one summe, they sette not 5 at the left hande of the scores of pounds, but they set all the hundreds in an other far∣ther rowe, and 500 at the lefte hande thereof, and the thousandes they set in a farther rowe yet, and at the lefte side thereof they sette the 5000, and in the space ouer they sette the 10000, and in a higher rowe 20000, whiche all I haue expressed in this example, whyche is 97869 lb, 12 s, 9 d, ob, q. Ninetye seauen

thousande, eighte hundred, thrée score & myne pounde, twelue shillings and nine pence half∣peny farthing, for I had not tolde you before, where, neither how you should set down far∣things, which (as you sée here)

[illustration]

must be set••e in a voide space sideling beneath the pens: for q one counter, for ob,* 1.15 2 coun∣ters: for ob, q, 3 counters: and more there cannot be: for 4 farthings make 1 d, which must be set in his due place.

And if you desire the same summe after Auditours man∣ner: Lo here it is.

But in this thing you shall take this for sufficient, and the rest you shal obserue as you may sée by the working of ech sort: for the di∣uerse wits of men haue inuented diuerse and sundrie wayes, almost vnnumerable.

THE ARTE OF NVM∣bring on the hande.

BVt one feate I shall teach you, which not onely for y^e strange∣nesse and secretenesse is muche pleasant, but also for the good commoditie of it, right worthy to be wel marked.

This feate hath bin vsed aboue 2000 yeres at the least, and yet was it neuer commonlie knowen, especiallie in Englishe it was neuer taught yet. This is the arte of numbring on the hand, with diuerse gestures of y^e fingers, expressing anye sum conceiued in the minde. And first to beginne.

If you will expresse anye sum vnder 100, you shall expresse it with your left hand and from 100 vnto 1000, you shal expresse it with your right hand, as here or∣derly by this Table follo∣wing you maye perceiue.

Here followeth the Table of the Art of the hande.

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[ 1] 1 In which (as you may sée) 1 is expressed by the little finger of the left hand, closely and harde crooked.

[ 2] 2 Is declared by like bowing of the wed∣ding finger (which is the next to the little fin∣ger) togither with the little finger.

[ 3] 3 Is signifyed by the middle finger, bowed in like manner with these two.

[ 4] 4 Is declared by the bowing of the middle finger, and the ring finger or wedding finger, with the other all stretched forth.

[ 5] 5 Is represented by the middle finger one∣ly bowed.

[ 6] And 6 by the wedding finger onlie crooked: and thus you may mark in these a certain or∣der. But now 7, 8, and 9, are expressed wyth the bowing of the same fingers, as are 1, 2, & 3, but after another forme.

[ 7] For 7 is declared by the bowing of y^e little finger as is 1, saue that for 1 the finger is clasped in, hard and round, but for to expresse 7, you shall bow the middle ioynt of the little finger onelye, and holde the other ioyntes straight.

Sch.

If you wil giue me leaue to expresse it after my rude maner, thus I vnderstād your meaning: that one is expressed by crooking in

the little finger, like the head of a Bishoppes bagle: and 7 is declared by the same finger bowed like a gibbet.

M

So I perceiue you vnderstand it.

Then to expresse 8, you shall bowe after [ 8] the same maner both the little finger, and the ring finger.

And if you bowe likewise with them the [ 9] middle finger, then doth it betoken 9.

Nowe to expresse 10, you shal bowe your [ 10] forefinger rounde, and set the end of it on the highest iointe of the thumbe.

And for to expresse 20, you muste set youre [ 20] fingers straight, and the end of your thumb to the partition of the formost & middle finger.

30 Is represented by the ioyning togither [ 30] of the heads of the foremost finger & the thūb.

40 Is declared by setting of the thumbe [ 40] crossewaies on the formost finger.

50 Is signified by right stretching foorth of [ 50] the fingers iointly and applying of y^e thumbs ende to the partition of the middle finger, and the ring finger or wedding finger.

60 Is formed by bending of the thumbe [ 60] crooked, and crossing it with the forefinger.

70 Is expressed by the bowing of the fore∣most [ 70] finger and setting the ende of the thūbe

betwéene the 2 formost or highest ioints of it.

[ 80] 80 Is expressed by setting of the foremoste finger crossewayes on the thumbe, so that 80 differeth thus frō 40: for that 80, the forefin∣ger is set crossewayes on the thumbe and for 40 the thumb is set crosse ouer the forefinger.

[ 90] 90 Is signified by bending the forefinger, and setting the end of it in y^e innermost ioint of the thumbe, that is euen at the foote of it. And thus are al the nūbers ended vnder 1••0.

Sc.

In déed these be al he numbers from 1 to 10, & then all the tenthes within 100, but [ 11, 12, 13, 21, 22, 23,] this teacheth me not how to expresse 11, 12, 13, &c. 21, 22, 23, &c. and such like.

Ma.

You can little vnderstand, if you cā not doe that without teaching. What is 11? is it not 10 and 1? then expresse 10 as you were taught and 1 also, that is 11: and for 12 ex∣presse 10 and 2: for 23 set 20 and ••: and so for 68, you must make 6, and thereto 8: and so of al other sortes.

But now if you wold represent 100, either any number aboue it, you must doe that with the right hand, after this maner.

[ 100] You must expresse 100 in the righte hande with the little finger, so bowed as you did ex∣presse 1 in the left hand.

And as you expressed 2 in the lefte hande, the same fashion in the right hande doeth de∣clare 200. [ 200]

The fourme of 3 in the right hand standeth for 300. [ 300]

The forme of 4 for 400. [ 400]

Likewise the forme of 5, for 500. [ 500]

The forme of 6, for 600. And to be shorte: [ 600] looke how you did expresse single vnities and tenthes in the left hande, so must you expresse vnities and tenthes of hundreds, in the right hand.

Scholer.

I vnderstande you thus: that if I woulde represente 900, I muste so forme the [ 900] fingers of my right hande to expresse 9. And as in my lefte hande I expresse 10, so in my right hand must I expresse a 1000. [ 1000]

And so the forme of euerie tenth in the lefte hand, serueth to expresse the number of thou∣sands, so the summe of 40 standeth for 4000. [ 4000]

The summe of 80, for 8000. [ 8000]

And the forme of 90 (which is the greatest) for 9000, and aboue that I can not expresse [ 9000] any number.

Mayster.

No, not with one finger, howe be it, with diuerse fingers you maye expresse 9999. and al at one time, & that lacketh but

1 of 10000. So that vnder ten thousande you maye by your fingers expresse anye summe. And this shall suffise for Numeration on the fingers. And as for Addition, Subtraction, Multiplication, and Diuision (whiche yet were neuer taught by any man as farre as I doe knowe) I will instruct you after the trea∣tise of Fractions: and now for this time fare well, and looke that you ceasse not to practise that you haue learned.

Sc.

Sir, with most hartie minde I thanke you, both for your good learning and also your good counsel, which (God willing) I truste to follow.

FINIS.

¶THE SECONDE part of the Arithmetike touching Fractions, brief∣lie set forth.

Scholer.

ALbeit I perceiue your manifolde businesse doth so occupie, or ra∣ther oppresse you, that you can not as yet cō∣pletelie end that trea∣tise of Fractions A∣rithmeticall, whyche you haue prepared, wherein not onelie sun∣drie workes of Geometrie, Musicke, and A∣stronomie be largelie set forthe, but also di∣uers conclusions and naturall woorkes, tou∣ching mixtures of metals, and compositions of medicines, with other straunge examples,

yet in the mean season. I can not stay my ear∣nest desire, but importunelie craue of you some briefe preparation, towarde the vse of Fractions, whereby at the leaste I maye be able to vnderstande the common workes of them, and the vulgare vse of those rules, whi∣che without them can not wel be wrought.

Maister.

If my leasure were as great as my will is good, you shoulde not néede to vse anye importunate crauing, for y^e attaining of that thing, whereby I maye be perswaded that I shal anye waies profite the common wealth, or helpe the honest studies of anye good mem∣bers in the same: wherefore, while myne at∣tendaunce will permitte me to walke and talke, I am well willing to helpe you as I may.

* 1.16Therefore firste to beginne with explica∣tion of this name Fraction, what take you if to be?

Scholer.

Marie sir, I thinke a Fraction (as I haue heard it often named) to be a broken number, that is to say, to be no whole nūber, but a part of a number.

Mayster

A Fraction in déede is a broken number, and so consequentlie, the part of an∣other number: but that muste be vnderstan∣ded

of suche an other number, as can not bée diuided into any other partes thā Fractions: for although I may take the third part of 60, or the fourth part of it, and so of other partes diuerslie, yet these partes be not properlie, nor ought not to be called Fractions, bycause they maye be expressed by whole numbers: for the third part of it is 20: the fourth part is 15: the twelfth part is 5, and so forth of other parts, which all be whole numbers.

Wherefore properlie a Fraction expresseth the partes or part onelie of an vnit,* 1.17 that is to saye, that the number which is the whole or entire summe of anye Fraction, may not be greater than one: and therefore it followeth, that no one Fraction alone can be so gret, that it shall make 1, as by examples I will de∣clare as soone as I haue taught you to knowe the forme howe a Fraction is expressed or re∣presented in writing.

NVMERATION.

BVt first to begin with the expressing of a Fraction, which is the numeration of it, you must vnderstand that a Fraction is repre∣sented by 2 numbers, set one ouer the other, and a line drawen betwéene them as thus, ⅓.••/4.⅘ 10/17 which foure Fractions you muste pro∣nounce thus: ••/3 one thirde parte: ¾▪ thrée quar∣ters: ⅖ two fifte partes: 10/17. tenne seauentéene partes.

Scho.

I vnderstande the forme of theyr ex∣pression and pronunciation, but their mea∣ning or valuation séemeth more obscure: yet I think that by the two first Fractions I vn∣derstand the valuation of the two later Fra∣ctions, and so consequentlie of other.

M.

Value them then, that I may perceyue your taking of them

Scholer.

⅖ betokeneth two fifte partes, that is to say, if one be diuided into 5 parts, that Fraction doeth expresse ij. of those fifthe

partes: 10/17 doth signifie, that if one be diuided into xvij. partes, I must take tenne of them. And this I gather of the two firste examples: for ⅓. that is one thirde parte, doth easily de∣clare, that if anye one thing be diuided into three partes, I muste take but one of them: so ¾ that is thrée quarters, doeth declare that one being diuided into four quarters, I must take (for this Fractiō) thrée of those quarters.

If there be no more difficultie in their Nu∣meration, thē I pray you go forward to their Addition and Subtraction,* 1.18 and so to the other kinds of workes for I vnderstande that the same kinds of workes be in Fractions, that be in whole numbers.

Maister.

There are the same kynds of workes in bothe, albeit the order of them is diuerse, as I will anone declare: but yet more in Numeration before we leaue it. You muste vnderstande, that those two numbers whiche expresse a Fraction, haue seuerall names.* 1.19 The ouermost whiche is aboue the line, is called the Numerator, and the other beneath the lyne, is called the De∣nominator.

Scholer.

And what is the reason of theyr diuerse names? For in mine opinion both bee

Numerators, séeing both they do expresse the numeration of the Fraction.

Maister.

You are deceiued: for one onelye (whiche is the ouermoste) doeth expresse the Numeration: and the denominatour doeth declare the number of partes into whiche the vnit is diuided, as in this example, when I say: Diuide a pounde weighte of Golde be∣twéene foure men, so that the firste man shal haue 2/15 the seconde 2/15 the thirde 4/15 and the fourth 6/15.

Now do you perceiue y^e by the denominator (whiche is one in al foure Fractions) it is in∣tended, that the pounde waight shoulde be di∣uided into so manye partes I meane 15, and by the foure seuerall numerators is limitted the diuerse portion that each man shold haue, that is, that whē the whole is parted into 15, the firste man shall haue 2 of those 15 partes: the second man thrée of them: the third man 4: and the fourth man 6. And so may you sée y^e seueral offices (as it were) of those two num∣bers, I meane of the Numerator and the de∣nominator.

And hereby you perceiue, that a man can haue no more parts of any thing than it was diuided into, nether yet aptlie so many: so that

it were vnaptly sayd: You shall haue 15/15. that is xv fiftéene partes of any thing, séeing it were better sayde: You shal haue the whole thing.

Sc.

So doth it appeare reasonablye: for the labour is vaine, to diuide anye thing, and than to applie the Diuision to no vse. And much lesse reasonable were it to say 16/15: for if the whole be diuided into 15 parts on∣ly, it is not possible to take 16 of them, that is to say, more than altogither.

Maister.

This is true touching the proper and apte vse of the name of a Fraction:* 1.20 yet improperlye, and after a vulgare acceptation (for easinesse in worke) both those formes be called Fractions, because they be writtē like fractions, although they be none in déede for 15/15, and generally all suche other: where the Numerator and Denominator be equal, are not Fractions: but the whole thing with all his partes. And so 16/12 is not to be called a fra∣ction, but a mixt number, of a whole number and a Fraction: for it is as muche, as 1 4/12, that is one whole one, and 4 twelue partes, as shall be declared in Reduction. Therefore they doe abuse the names, that

call them Fractions, where the Numerator is either equall or greater than the Deno∣minator.

Sc.

But is there any néedefull cause why they should so abuse the name?

Mai.

There is cause why they shal some∣times, for easinesse in worke, write some nū∣bers after that sorte, like fractions: but they néeded not to call them fractions, but as they be whole numbers or mixt numbers (that is whole numbers with Fractions) expressed like fractions.

Nowe must you vnderstande, that as no fraction properly can be greater than 1, so in smalnesse vnder one the nature of Fractions doeth extende infinitelye: as the nature of whole numbers is to increase aboue one infi∣nitelye, so that not onely one, may be diuided into infinite Fractions or parts, but also eue∣rye Fraction maye be diuided into infinite Fractions or partes, whiche commonlye bee called Fractions of Fractions, and they be expressed diuerslye: As for example, 3/••.⅔.½, that is thrée quarters of two third partes, of one halfe parte. Whereby is signifyed, that if one be diuided into two halfes, and the one halfe into thrée partes, and two of

those thrée partes, be diuided idyntlye into foure quarters, this Fraction of Fracti∣ons doeth represente three of those quar∣ters.

Scholer.

I praye you lette me prooue by an example in common money, whether I doe rightlye vnderstand you or no. One Crowne, which I take for an vnit, doth con∣taine 60 pennies, therefore the halfe of it is thirtye pence: ⅔ of that halfe is 20 pence, whereof 3/•• is fiftéene pence, so then 15 pence is 3/••.2/••.½ of a Crowne. And so 3 pence is ¾.⅔.½. of a shilling.

Mayster.

You perceyue this well y∣nough, but how happened that you founde no doubte in the forme of writing these Fra∣ctions, séeing the two latter Fractions haue no line betwéene their numbers, as the firste hath?

Sc.

Because I had forgotten (as Scho∣lers oft times doe) that that was tolde me before: but I praye you, expresse the reason thereof.

Mai.

This forme is but voluntarye, and therefore hath none other reason than the will of the diuiser, which forme many do fol∣low. Some other doe make lines betwéene e∣uery

Fraction, and adde wordes of distincti∣on, after this sorte, ¼ of ⅔, of ½, which forme is good also.

Some other expresse them thus 〈 math 〉〈 math 〉 in slope forme, to distincte them from seuerall Fractions of one whole number, for if they were set in one right line thus,* 1.21 ¾ ⅔ ½. then oughte it to be pronounced, thrée quarters, and two thirde partes and an halfe, which maketh al∣most two whole vnits, lacking but one xij. part. And so is it nothing agréeable with the other Fraction of Fractions, wherefore it is a great ouersight in certayne learned men, which doe expresse them so confusedly with such seuerall Fractions, that a man can not know the one from the other.

Therefore some men (as Stifelius) doe expresse without a line numbers of proporti∣on, being applied to Addition or Subtracti∣on: because they must be takē as two, where the line in Fractions maketh them to be ta∣ken for one: for of the Numeratour and De∣nominatour is made one number.

Scholer.

* 1.22Then I perceiue there be thrée seuerall varieties in Fractions: First when one only Fractiō is set for one nūber, as 4/••, y^t

is foure fifth parts. The second, is whē there be set two or more seuerall Fractions of one number, as ⅘ 2/5, that is iiij. ninth partes, and two fift parts. The third sort is Fractions of Fractions, as 4/9 2/5, that is 4 ninth parts of two fifth parts.

Ma.

You haue said well, if you vnderstand well your own wordes.

Scho.

If it shall please you, I will by an example in the parts of an old Englishe An∣gell expresse my meaning.

Mai.

Let me heare you.

Scholer.

The olde Englishe Angell did containe 7 shillings 6 d, that is 90 d. Nowe ⅘ of it, is 72 d. And of the same 90 pence, if I take ⅘ and ⅖, that is foure ninth partes, and 2 fifth partes, 4/9 is 40, and ⅖ is 36, which both make 76: but if I take 4/9 of ⅖, that is foure ninth partes of two fift partes, séeing ⅖ is but 36, then 4/9 of 36 will yield but 16: for 1/9 of 36, is but 4, and that taken foure times maketh 16.

Maister.

This is plainely expressed, and truely and hereby (I doubte not) but you doe perceiue, y^t as great a differēce as is betwéen 16 & 76, so much differēce is betwéene these two Fractions 4/9 and ⅖: and 4/9 of ⅖.

And now that you vnderstande these varie∣ties, I will procéede to the rest of the works: first admonishing you, that there is an other order to be followed in Fractions than there was in whole numbers, for in whole num∣bers this was the order: Numeration, Addi∣tion, Subtraction, Multiplication, Diuision, and Reduction, but in Fractions (to followe the same aptnesse in procéeding from the easi∣est workes to the harder) we must vse this order of the workes: Numeration, Multipli∣cation, Diuision, Reduction, Addition, and Subtraction.

Scho.

That Multiplication and Diuision shoulde goe togither, and Subtraction to followe Addition, naturall order doeth per∣swade: but why Multiplication shoulde be first in order here next to Numeration, and Reduction in the middle, I desire to vnder∣stand the reason

May.

As in the Arte of whole numbers order woulde reasonablye beginne with the easiest, and so goe forwarde by degrées to the hardest, euen so reason teacheth in Fractions the like order. And considering that Addition or Subtraction of Fractions can very seldom be wrought without multiplication and Re∣duction:

and contrariwayes, Multiplication and Reductiō may be wrought without this forme of Addition or Subtraction. Therfore was it orderly required, that Multiplication and Reduction shoulde goe before Addition and Subtraction. And the same reason ser∣ueth for the placing of Multiplication before Reduction.

Sch.

Then if Multiplication be the easi∣est, I pray you declare the forme of it first by rule, and then by example.

May.

Your example is good.

MVLTIPLICATION.

THerefore when anye two Fractions be pro∣poned to be multipli∣ed togither, y^e Nume∣rator of the one must be multiplied by the Numerator of the o∣ther: and the summe that amounteth therof, must be set for a new numerator: likewise the Denominatour of the one must be multiplied by the Denomi∣nator

of the other, and that that amounteth, shal be set for the common Denominator: & this new third Fraction expresseth the Pro∣duct of the multiplication of the two first fra∣ctions proponed, whereof take this example, ⅗ multiplied by 5/12, doth make 15/6••.

Scholer.

I perceiue then, that 3 being the Numeratour of the first Fraction, is multi∣plied by 5, being the Numeratour of the se∣conde Fraction, whereof amounteth 15, the Numeratour of the thirde Fraction. And so likewise, 5 being Denominatour of the first Fraction, is multiplied by 12 the Denominator of the second fraction, where∣of amounteth 60 the newe Denominator: so that I perceiue howe the worke is done, but I doe not perceiue howe 15/60 is greater than ⅖: For if I shall vse my former maner of examination by the partes of some Coyne, I sée that ⅗ of a Crowne is 36 d, and 5/12 of a Crowne, is 25 d, whereof the one multiply∣ed by the other, doth make 900 d, whiche is 15 Crownes: but by your multiplication there amounteth ••••/60, which is but 15 d, and that is much lesse then anye of both the firste Fractions.

Mayst.

That difference is betwéen mul∣tiplication

in whole numbers, and multipli∣cation in broken numbers that in whole nū∣bers the summe that amounteth, is greater than both the other whereof it came: but in Fractions it is contrarywaies: for the sum that amounteth is lesser thā any of the other two fractions, whereof it came.

Sc.

I desire much to vnderstande the rea∣son thereof.

May.

Although I purposed to reserue the reasons of workes Arithmeticall for the per∣fecte Booke of Arithmetike, yet I will shew you this, because of the straungenesse of the worke.

You sée in whole numbers, that of two numbers being multiplied together, is made the third number: which third number doth beare the same proportion to the nūber mul∣tiplyed, that the multiplier doth beare to an vnit. And so in Fractions, the third number which amounteth of multiplication, beareth the same proportion to eache of the two firste fractions, that the other of those two fractiōs doth beare to an vnit.

Scholer.

Sir I vnderstande your wordes thus: when 40 is multiplied by 12, there doth amount 480, whiche 480 doeth con∣tayne

40 so many times in it, as twelue doth contayne vnits, that is to say: twelue times. And so it appeareth, that 480 doeth contain twelue so many times also, as 40 doeth con∣tayne Vnites, that is 40 times. But now I see not how the thirde number in this exāple of Fractions can contain any of the two for∣mer (as it happened in whole nūbers) séeing it is lesser than eyther of them.

Mai.

No maruaile if you cannot sée that thing which is not possible to be séene of anye man, how the third number in multiplicati∣on of Fractions should be greater than anye of the two former Fractions, but yet this may you sée (which I sayde) that the thirde number in Fractions so multiplyed, doeth beare the same proportion to any of the two former fractions, that the other of those 2 fra∣ctions doth beare to an vnite, as in your ex∣ample ⅗ being multiplied by 5/12, doeth make 15/60. Now saye I, that 15/60 doeth beare the same proportion to ⅗, that 5/12 doth beare to an vnit, as you may in your own forme of examina∣tion by coyne trie it. For in an olde Angell are 180 halfe pence, whiche I set for the in∣tire vnit whose parts (according to the Fra∣ctions aforesayde) are these, for 15/60 sette 45 ob.

for ⅖ take 160 ob and for 5/12 put 75 ob Nowe doeth 45 beare the same proportion to 108 that 75 doeth beare to 180: for 45 is 5/11 of 108, and so is 75 also 5/12 of 180. And for easi∣er applying of ech comparison, consider this forme of setting all these nūbers before your eyes, where the secōd demōstration towards your right hand is answerable to the firste in euerye proponed part, where for ⅗ (of 180) stands his value 108: for ••/•• stands 75: and for 15/60 is 45.

〈 math 〉〈 math 〉

But these reasons may be better reserued till another time, whē the knowledge of pro∣portions in due order shall be taught. Yet in the meane season I will shewe you howe it commeth to passe that in Fractions the third summe must néedes be lesser then any of the other two.

Consider thus, that when a Fraction is pro∣poned, as in the former example ⅗, if it bée multiplied by more thā 1, it will make more

than one entire number. As if I multiplie by 5, that is to saye if I take it 5 tymes, it will make thrée entire vnits: example in a Crowne, ⅗ of it maketh 3 s, which if I take fiue times, it will amounte to 15 shillinges, that is thrée intire Crownes: so if I take the same ⅗ but twice, it will yielde 6 s, that is one entire Crowne and ⅕. Now if I take it but once it cannot be more thā it was before, that is,* 1.23 3 s. And if I take it lesse then once, it cannot be so much as it was before. Then seeing that a Fraction is lesse than one, if I multiplie a Fraction by another Fraction, it followeth that I doe take that first Fraction lesse than once and therfore the summe that amounteth, must néedes be lesse than the first fraction.

Sc.

Sir, I thanke you much for this rea∣son, And I truste I doe perceiue the thing, as by example of this same Fraction ⅗ I wil expresse If I take ⅗ of a Crowne once, that is to say, if I multiplie ⅗ by 1, it will be as it was before, but 3 s: so if I doe multiplie it by ½, that is, if I take it but half one time, then will it be but halfe so muche: likewise if I multiplie it by ½, that is, if I take but the thirde part of once, it will yielde but 12

pence, that is the thirde part of the first Fra∣ction.

And so to make an ende. If I take it but the twelfth part of once, that is, if I doe mul∣tiplie it by ••/••••, it will yielde but the twelfth part of the firste Fraction, which is but thrée pence. And it followeth that if 1/••2 make 3 pence, then 5/12 must néedes make fiue tymes so muche, that is 15 pence, which was the summe that hath giuen the occasion of all this doubt.

Maister.

Then I perceiue you haue suf∣ficient vnderstanding in this sorte of multi∣plication for this time, wherefore I will o∣mitte that I might saye more of Multiplica∣tion, till we come to reduction, and wil passe to the other workes, and firste to Diuision, whose place followeth Multiplication, both by naturall order, and also in eāsinesse of worke.

DIVISION.

WHen so euer two fra∣ctions bée proponed, that the one shoulde be diuided by the o∣ther, I muste sette downe firste the Fra∣ction that shall be di∣uided (which is called the Diuidend) and then after it the other, whiche is the Diuisor. Then shall I mul∣tiplie the numerator of the diuidend by the denominator of the diuisour, and that which amositeth, I must put for a new numerator. Agayne, I shall multiplie the denominator of the diuidend by the numerator of the di∣uisor, and the number that amounteth there∣of, I must put for the new denominator. And this third fraction is the quotient of the sayd diuision.

Scholer.

This séemeth easie in forme, as by example, thus: If I would diuide ⅝ by 2/6, first I must multiplie 5 (being the numera∣tor of the diuidend) by 6, whiche is the deno∣minator

of the Diuisor, and thereof riseth 30: then I multiplie 8 (being the denomi∣nator of the diuidend) by 2, being Nu∣merator in the diuisor, and so riseth 16, the whiche I muste make in a thirde Fraction, thus 30/16.

Ma.

Me séemeth you are quicker in vn∣derstanding nowe, than you were when I taught you the arte of whole numbers: but that is no maruell, for the more knowledge that any man getteth, the readier shal he find his wit, and quicker in vnderstanding: but yet of 2 thinges I will admonish you, which you mighte haue obserued here for ease of worke and lightnesse of vnderstanding the nature of the Quotient.

Whensouer you diuide one Fraction by an other, either they be both equall togither eyther else the one is greater than the other: if they be equall, their quotient shal be such, that the numerator and the denominator of it shall be equall also. And if y^e 2 firste fractions be vnequall, their quotient shall declare the same by the vnequalitie of the numeratour and denominatour, as in these examples fol∣lowing shall appeare.

First of equall Fractions: 4/9 and 12/2•• bée e∣quall togither: and if the one be diuided by the other, the quotient will be 108/108, as you may perceiue by that rule aforesayde.

Now in the vnequal Fractions, as 4/9 and 3/10 the quotient wil be 40/27: where the Numera∣tor is greater than the denominator.

Sch.

I sée it is so, but I see not the reason why it should be so.

Maister.

* 1.24The reason is this, when anye Fraction is diuided by an other, the quotient declareth what proportion the diuidend bea∣reth to the diuisor. So ½ diuided by ¼, ma∣keth 2, which must be sounded, not two, but twice: declaring that ¼ is contained twice in ½.

And note this, that the Numerator in the Quotient, representeth the Diuidend, & the Denominator representeth the diuisor▪ And this is alwayes true,* 1.25 whether, the greater fraction be diuided by the lesser, or the lesser by the greater. But this proportion will not be exactly knowne, till you haue learned the art of proportions: notwithstāding somwhat of it will I declare in the next rule of Redu∣ction. But now for the easie remembrance of the Quotiēt in diuision, as soone as you haue

set downe your two Fractions, the one a∣gainste the other, then make a streighte line for the quotiente: and as soone as you haue multiplyed the Numerator of the diuidende, by the Denominator of the diuisor sette the number that amounteth, ouer the said line, and then multiply the other two numbers, & set their totall vnder the same line.

Scholer.

I perceiue you woulde not haue me trust to memorie til I wer better expert, least oftentimes I happen by misse remem∣braunce to be abused. This example I take for that declaration.

If I woulde diuide ⅔ by ¾ I must set the numbers one against the other, 〈 math 〉〈 math 〉 (as here doeth appeare) & then make an other line for the Quotient in some good distaunce, where I maye set the num∣bers of the Quotient, as soone as any of them is multiplied: So then as soone as I haue multiplied 2 by 4, which maketh 8, 〈 math 〉〈 math 〉 I shal set that ouer that line thus. And then multiplye •• by ••, whiche yéeldeth 9: and that 9 muste I set vnder the same line, and then will the whole quotient appeare thus 8/9.* 1.26 Whereby appeareth (as I remember your wordes) that ⅔ is in propor∣tion

to ¼, as 8 is to 9: but howe maye I per∣ceiue that?

Mayster.

Althoughe you shall better per∣ceiue it by the rule of Reduction, yet this ex∣ample may be declared in common coine, as in a common shilling of xij. pence, of whiche ••/2 maketh 8 d, and ¾ dothe make 9 pence, and so you maye easily see that their proportions doe agrée. And if you had taken this example before, when you tooke the example of▪ and ••/6, your Quotiēt would appere (as this doth) more easier to vnderstande, whereas that Quotient being 10/16, is not an easie proportion for you to perceiue, being yet little acquain∣ted with proportions: whereof to giue you some taste, I will enter to the rule of Redu∣ction: in which also I wil declare other wor∣kes, both of Multiplication and also of Di∣uision, whiche nowe I muste for a time o∣mitte, as things that do neede the help of Re∣duction.

REDVCTION.

THerefore wil I nowe declare the diuersities of Reduction of Fra∣ctions,* 1.27 whiche com∣monly haue fiue vari∣eties.

1. First, when there be, sundrie Fractions of one entire Vnitie, they muste be reduced to one denomination, and also into one Fra∣ction.

2 Secondarily, when ther be proponed fra∣ctions of fractions, they must be reduced like∣wise into one Fraction, for other wayes they can not bée brought into one Denominati∣on.

3 Thirdly, when an Improper fraction is proponed, that is to say, a fraction in forme, which indéed is greater thā an vnitie, it must be reduced into apt form expressing y^e Vnitie or Vnities of it, & the proper fraction distinct∣ly. And some times also it shall be néedefull to conuerte such a mixte number of Vnities,

with Fractions into the forme of a Fracti∣on, that is into an Improper Fraction, which 2 formes I esteeme but as one, bicause they worke on one kind of number.

4 Fourthlie, there happeneth sometimes Fractions to be written in greate num∣bers, whiche mighte be written in lesser numbers, therefore is there a meane to re∣duce suche greate numbers into their smal∣lest tearmes.

5 Fiftelie, when anye Fraction betoke∣neth the partes of a whole thing, whyche hath by common partition certaine partes, but none of like Denomination wyth that Fraction, then maye you reduce the sayde Fraction into an other, whose Denomina∣tion shall expresse the common partes of that whole thing.

Scholer.

Thys distinction in doctrine de∣lighteth me much, but more with hope than presente fruite, for as yet I doe not vnder∣stande scarselie the varieties, and much losse the practise and vse of their workes.

Mayster.

Reduction is an orderlie al∣teration of numbers out of one forme in∣to an other, whiche is neuer done order∣lye but for some néedefull vse, as in euery of

the saide 5 seuerall varieties I wil distinctly declare.

Firste therfore, when two or more seue∣rall Fractions of anye Vnite be proponed,* 1.28 as for example, ••3/1 and 4/6: because it is harde to tell what proportion of the entier num∣ber those two Fractions doe expresse, there∣fore was Reduction diuised, to bée a meane whereby these seuerall Fractions mighte be broughte into one Denomination and Fraction.

And in these Fractions this is the arte for bringing them to one denomination.

Multiplie firste the Denominators to∣gither,* 1.29 and the totall thereof you shall sette twice downe vnder two seuerall lines for two newe Denominators, or rather for one common Denominator: Then multiplye the Numeratour of the firste Fraction, by the Denominator of the seconde, and sette the totall thereof for the Numeratour ouer the firste line. Likewise multiplie the Nu∣merator of the seconde Fraction by the De∣nominator of the first, and set that totall ouer the seconde line for the Numerator of that Fraction, and so are those two first Fractions of seuerall denominations, brought to one

Denomination.

Sch.

If I vnderstand you, as I thinke I do, my example shal declare the same. The Fra∣ctions which you proponed were these, ••/16 and 4/6▪ whose Denominators (being 16 and 6) I multiplie togither, and there amounteth 96, which I set vnder lines, thus. 〈 math 〉〈 math 〉 Then I multiplie the Numerator of the first Fraction by the Denominator of the second, saying: 3 into 6 maketh 18, that set I ouer the first line for a newe Numeratour, and it wil be thus. 〈 math 〉〈 math 〉

Likewaies I multiplie the Numeratour of the second Fraction by the Denominatour of the first: saying: 4 times 16 maketh 64, that I set for y^e second Numeratour, and the Fra∣ction wil appeare thus. 〈 math 〉〈 math 〉 So that both Fractions broughte to one De∣nomination, must stand thus: 〈 math 〉〈 math 〉

Ma.

You haue done wel.

Scholer.

I beséech you, let me examine it after my accustomed forme, by common parts of coine.

Ma.

Go to.

Scholer.

A new Angell accompted at eight shillings, contayneth 96 pence, whereof ••▪

that is the xvj. parte, is sixe pence, and 1/16 is 18 pence, that is 18/••6. Againe ⅙ of the same An∣gel, is 16 pence, so that 4/6 maketh 64 d, that is 64/••••. And so I find the summes to agrée with the other before.

Maister.

So haue you nowe the Arte to bring such two Fractions into one Denomi∣nation. And if there be more than ij,* 1.30 then must you multiplie al the Denominators togither, and set the totall therof so many times down as there be Fractions, and then to get for ech one a newe Numerator. Multiplie the Nu∣merator of the firste, by the Denominator of the second, and the total therof multiplie by y^e denominator of the third, and so foorth if there be more. Likewise multiplie the Numerator of the second, by the denominator of the first, and the total thereof by the Denominator of the thirde. And in the same sorte multiply the Numerator of the third into y^e Denominator of the first: & the totall thereof into the Deno∣minator of the seconde, and so foorth, if there were mo. So these 3 Fractions 2/5 ••/4 2/•• doeth make by Reduction these other 3. Fracti∣ons of one Denomination 24/6•• 45/6•• 4••/6••. All whiche you may bring into one Fraction by

adding the Numeratours togither, and put∣ting that totall for the common Numerator, reseruing still that same common Denomi∣nator, And those 3 Fractions make one Im∣proper Fraction thus. 〈 math 〉〈 math 〉

Scho.

All this I perceiue, and also that thys laste Fraction is more than an Vnitie, and therefore you did call it an Improper Fra∣ction.

M

There be certaine other formes of wor∣king in this reduction, whiche I will brieflie touch also, to giue you an occasion to exercise your wit therin.

* 1.31The firste varietie is this. When you haue made and written downe youre common Denominator (as I haue taught before) then to get a Numerator for the first, do thus. Di∣uide the common Denominator by the De∣nominator of the first Fraction, and the quo∣tiente multiplied by the Numerator of the same, yéeldeth a new Numerator for the first new Fraction. So likewise doe with the se∣cond and the third, and with all the residue if there be more.

Scholer.

That will I proue in your laste example of these 3 Fractions 〈 math 〉〈 math 〉. When the Denominators be multiplyed, they make

60 for 5 into 4 maketh 20, and 20 by 3 yel∣deth 60, that I sette downe 3 times, thus. 〈 math 〉〈 math 〉: then to haue a Numerator for the firste, I muste diuide 60 by 5, (the Denomi∣nator of the firste) & the quotient is 12, which I muste multiplye by 2 (the Numerator of the firste) & that maketh 24, and so haue I for the first Fraction 24/60.

Likewyse for the seconde fraction:* 1.32 I diuide 60 by 4, and there commeth 15, whiche I multiplye by 3, and so haue I 45 and the se∣conde fraction 45/60. Then for the third in like sort wil come 40/60.

Mayster.

An other way is this. If it hap∣pen so that the lesser Denominator can by a∣ny multiplication make the greater, thē note the multiplier, and by it multiplie y^e Nume∣ratour ouer that lesser Denominatour, and for the lesser Denominatour put the greater, as thus in these two Fractions, 2/••2 and ⅔ thrée being the lesser Denominatour multiplyed by 4, will make 12, whyche is the greater Denominatour: therefore by the same 4, I do multiplie 2, whiche is the Numeratour ouer 3, and that maketh 8: vnder which I doe put 12 being the greater Denominatour, whiche is also made by Multiplication of 4 into ••, &

so haue I these ij. Fractiōs 3/••2••, 8/••2: thus short∣lye reduced without altering the one Fra∣ction.

S

This I vnderstande.

Ma.

* 1.33Then marke this thirde waye: If the denominators doe not happen so, that one by Multiplication may make the other, thē loke whether they both maye be partes of anye o∣ther one number, as in ••/1•• and ••/18, althoughe the lesser taken but twice be to great to make 18, yet they both may be parts vnto 36: ther∣fore loke howe manie times 12 is in 36, and that quotient being multiplyed by the nume∣ratour ouer 12, the totall shall be putte in steade of the Numerator ouer 12, & for 12 put 36, thus, ••5/••6. So likewise looke howe often is 18 in ••6, and because it is twice, therefore by 2 multiplie 7. whiche is ouer 18, and it wyll be 14, set that for the numerator, and in stead of 18 put 36, and then shall youre Fractions reduced, stande thus, 1••/36 14/36 in steade of 5/••2 and 7/1••.

* 1.34And if you will proue whether you haue wrought well or no, that maye be proued by Reduction of them againe to their former de∣nominations, whiche arte shall be taughte in the fourth kinde of Reduction, where greater

termes of Fractions be reduced into smaller in number, but no smaller in proportiō.* 1.35 And if in suche Reduction the same termes or nū∣bers come again that were before, then is the worke good, else not.

S

Sir, I heare your wordes, but I doe not vnderstand many of thē, which it may please you to declare.

Ma.

With a good will, when conueniente place serueth, but that must be in the said iiij. kinde of Reduction. In the meane season I wil declare the seconde forme of Reduction, which teacheth howe to reduce Fractions of Fractions into one Fraction, and so to one Denomination.

When fractions of fractions be proponed,* 1.36 you shall multiplie the Numeratours of each into other, and set the totall for the new Nu∣merator, and then multiplie all the Denomi∣natours likewaies, and take their totall for the new denominator, and so are they speedi∣ly reduced.

Sc.

If that be al then I vnderstand it alrea∣die, as by this example I wil declare. These be the Fractions, ¼ of ⅔ of 6/7 of 7/9 which I wold reduce to one denomination.

Therfore begin I with the Numeratours,

and multiple them all togither, saying: 3 into 2 maketh 6, and 6 by 6 maketh 36, whiche multiplyed by 7, yéeldeth 352, that 〈 math 〉〈 math 〉 I set ouer a line for the Numera∣tor, thus:

Then I multiplie the denominatours, 4 by 3 maketh 12, & that by 7 bringeth 84, whiche multiplyed by 9, yéeldeth 756, the new deno∣minatour. And so the whole 〈 math 〉〈 math 〉 reduced fraction is this, which is to hard a Fraction for me to vnderstand yet.

Mai

You thinke so, and no maruaile, but a∣none you shal learn to iudge it easilie, for this Fraction is no more in déede then ⅓ although it be in greater tearmes, and therefore more stranger and more obscure.

And this sufficeth for this Reduction, saue that I will shew you by a figure of measure, the iuste rate and reason of this kinde of Fra∣ctions, and also the due vnderstanding of the Reduction.

The entier measure parted into 9.

1	2	3	4	5	6	7	8	9
1	2	3	4	5	6	7	7/9
.1.	.2.	.3.	6/7
1	2	3	4	⅔
1	2	3	¾

Here you sée the longest measure, (whyche standeth for the whole and entier quantitie) firste parted into 9 Diuisions, whereof 7 are seuered by the second measure: and thereof a∣gaine are parted out 6. And that 6 being di∣stinct into 3 partes, 2 of them are parted by the fourth measure, of which fourth measure, being diuided into 4 partes, the lowest mea∣sure doth containe ¾, so that the same ¾ muste be named, not 3/4 of the whole measure, but in déede is ¾ of ⅔ of 6/7 of 7/9 or as I woulde rather expresse it, 3/•• 1/•• 6/•• 7/••.

Scholer.

This example is so sensible, that I can not choose but sée it. And furthermore, I sée also, that the same fraction is equall to 3/9 of the entier measure, as the lines whiche runne vp and downe doe expresselie set forth.

Also I sée here, that ⅔.6/7, 7/9 is equall to 4/9 And further yet that 6/7.••/9 is equal to 6/5.

Ma.

I am gladde that you see it so wel, not doubting but you wil gather greater light of knowledge hereby.

* 1.37But nowe it is time that we come to the thirde forme of Reduction, which teacheth of Improper Fractions, that is to say, Mixt nū∣bers of Vnites and Fractions, although they appeare like Fractions, as this 2/•• 6/5, whyche doeth include 5 Vnites wholly, and ⅕ ouer. Wherfore first you shall know them, by that the Numeratour is greater than the Deno∣minatour.

Scho.

In déede sir that appeareth reasona∣ble, that if the Numerator doe expresse more partes to be taken of any vnitie then the De∣nominator doth signifie that vnitie to be diui∣ded into, it must néedes follow that such a fra∣ction importeth more than the whole, that is to saye, the whole with certaine partes ouer. But what Reduction is there in it?

Ma.

There be two seuerall kindes of Re∣duction, concerning suche Fractions. Some∣times it shal be néedefull to conuert those fra∣ctions into the Vnites and the proper fraction that wil remaine And sometimes contrarie

wayes it shal be mete to reduce mixt nūbers, that is vnities, written with fractions into the forme of one simple fraction, & so be there two wayes.

Sch.

What is the meane of the first waye to turne improper fractions into Vnities wyth their proper Fractions?

M.

That is thus.* 1.38 Your Numerator béeing greater than the Denominator, muste be di∣uided by the same denominator, & the quotiēt therof expresseth the vnities, y^e remainer shal be put for the Numerator of the fraction that resteth, and the denominator must be y^e same that was before.

Sc.

For example, I take 17/5. And diuiding 17 by 5, the quotient wil be 2, and there will remaine 2.

Ma.

That must you write thus, 3 ⅖ where (you sée) I haue written, without any line, as entier numbers ought to be writtē, and the 2 that remained, I haue set ouer the former denominator with a line, as a proper fractiō. And this number doth signifie now 3 Vnits, and ⅖ of one.

Scho

Then if I would by Vnites here vn∣derstand crowns, so it were 3 crownes, and ⅖, that is 2 s.

M.

Euen so, and therfore 17/5 did signifie the same. But this happeneth sometimes, that when the Reduction is so wrought, there re∣mayneth nothing. And then it is not a mixts number, but a simple entier number, repre∣sented like a Fraction.

Scholer

As ••••/•• wil make 3 iuste, and 15/3 wil make euen 6. This I wil remember. But nowe, what is the seconde forme of Reducti∣on, that you spake of for these sortes of Fra∣ctions?

M.

* 1.39Whensoeuer you haue any of these two sorts of numbers, that is to saye, whole num∣bers without Fractions, or whole numbers with fractions, and you would turne thē into the forme of a fraction you must multiply the whole number by that denominatour, which you wil haue to remaine stil, and to the total thereof adde the numeratour which you haue alreadie, and al that shall you set for the new numerator, kéeping still the former denomi∣natour: as if you haue 6 ¾ which you woulde conuert into an Improper Fractiō you must multiplie 6 by 4, whereof commeth 24, and thereto adde the numeratour which is 3, and so haue you 27 for the numerator, and 4 still for the Denominator.

Sc.

Then is 27/4 equall to 6 ¾.

Ma.

Euen iust, and so backward (as ap∣peareth by the former Reductiō) 6 ¾ maketh 27/4 And thus one of these Reductions may be the proofe of the others worke.

Sc.

This I perceiue, but now if you wold turne whole numbers without fractions in∣to any fraction, I sée not how that maye bée done, because there is no denominatour to make the multiplication by.

Ma.

That was well marked: but this you know, that no man intendeth to turne any whole number into a Fraction, but hée hath in his mind that denominator by which the Multiplication must be made:* 1.40 for the proofe whereof I sette downe 7, whiche is a whole number. And if you will haue this nū∣ber conuerted into any certayne fraction, will me to doe it.

Scholer.

I pray you reduce 7 into a Fra∣ction.

Ma.

Then you care not what the fraction be, so it be some fraction.

Sc.

No, I passe not for the sort of the Fra∣ction.

Ma.

Then how can you thinke that you require me to doe any thing certayne, when

you leaue me to doe as I list? and séeing you stande at that staye, whether thinke you that I must firste intende in mynde what fraction I will make of it, before I can doe it indeede?

Sc.

Else you should doe ignorauntly.

Mai.

Then I will limit my selfe (séeing you will not) to turne it into quarters. And therefore I multiplie 7 by 4 (whiche is the denomination of quarters) and there amoū∣teth 28 to be set for the Numeratour, and the 4 must be set for the Denominator, and the fraction will be thus, 28/••.

Sch.

Indéede I perceiue this to be reaso∣nable, for without much triall I vnderstand that 28/4 of any thing doeth make 7. And so then if I would turne 8 into fift parts, it wil make 40/5 which is all one with 8. for eyghte crownes turned into fift partes (that is, into shillinges) will make 40 shillinges, that is 40/5 of a crowne.

Ma.

Seing you vnderstande nowe these thrée kinds of Reduction, I will declare vnto you the fourth kinde, that is when Fractions be written in greater Tearmes than they néede, howe they maye be broughte to lesser Tearmes.

Scholer

To write anye thing in greater Tearmes than néedeth, seemeth to be a fault,* 1.41 and so this Rule séemeth to amende that fault.

Na.

It were a fault to do any thing with∣out néede which after must be redressed: but in this case it is not so: neither did I say ab∣solutely (as you doe) that it néedeth not to ex∣presse those Fractions in so greate tearmes, but that the fractions doe not neede, I meane for their value to be vnderstanded: but yet it maye be néedefull for the ease of those works whereto they be applyed, as for example: In the firste kinde of Reduction this was youre owne example: 3/16 and 4/••, whiche when you would reduce, you were fayne to turne them first into one Denomination, & so appeared they thus, 18/96, and 6••/96, where the Fracti∣ons (for their owne vnderstanding) néeded not to be turned out of smaller tearmes into greater, but yet the easinesse of working née∣ded it.

Sch.

Sir, I vnderstand nowe, not onelye the difference of this néede (for the Fractions mighte better be vnderstanded as Fracti∣ons seuerall, eche in his value,* 1.42 when they were in lesser tearmes, although they coulde

not so well be reduced) but also I vnder∣stand what you meane by greater Tearmes and lesser Tearmes, whereof before I was in doubt, for I see you call the Numeratour and denominatour, the Termes of the Fra∣ction.

Maister.

I am glad you vnderstande it so well. Now then when you woulde value a∣ny fractions (because that maye best be done when the Tearms are smallest) you shal re∣duce them to the smallest that you can, which thing you may doe thus:* 1.43 Diuide the greatest of any such two Termes by the lesser, and if any thing remayne by that remainer, diuide the last Diuisour: and if any thing remayne now, by that diuide the last diuisour (whiche was before the remainer of the first Diuisiō) and so continue still, till nothing do remayne in the Diuision: and then marke your laste Diuisour, for it is the number that will easi∣lye reduce your Fraction, if you diuide both the Numeratour and the Denominatour by the same number, and put for the Numera∣tour the quotient of his Diuision, and for the Denominatour also his quotient, that riseth by his Diuision.

Scho.

I take for example 18/96, and because

96 is the greatest number, I diuide it by 18, and the quotient is 5, and there resteth 6: what shall I doe with this quotient?

Maister.

Nothing in this worke, but nowe séeing there remaineth somewhat, by that remainer must you diuide the last diui∣sor.

Sc.

If I shal diuide 18 (which was the last diuisour by 6, that was the remayner, so is the quotient ••, and nothing resteth

Ma.

As for the quotient I omitte him yet: but because there doth remain nothing ther∣fore is 6 (which was your last diuisour) that number by which you may reduce the fracti∣on proponed,

Sch.

Then as you taught me. I muste diuide the Numerator 18 by 6 and the quo∣tient is 3, which I must put for 〈 math 〉〈 math 〉 the numerator ouer a line, thus. and then by the sayde 6, must I diuide also the denominator 96, and the quo∣tient will be 16, whiche I must take for the Denominator, and so is the fraction 3/16. And so me thinketh this Rule doth prooue y^e work of the first Reduction.

Maist.

That is true, if the first Reducti∣on were made of Fractions in their leaste

tearmes, and else not, withoute some helpe, as the second number in that place will de∣clare.

Sc.

The second nūber was ••/6, which was turned into 64/56, by that rule. Nowe if I shall by this Rule reduce it againe into the leaste tearmes, I must diuide 96 by 64, and there resteth 32, by which 32 I diuide 64, & there remayneth nothing, wherefore I must take that 32 for the Diuisour, to reduce the sayde Fraction. Then doe I diuide 64 by 32, and the Quotient is 2, whiche I sette for my Numeratour. Againe, I diuide 96 by 32, and the Quotient will be 3, and so haue I but ⅔

Ma.

Muse not at the matter, for you haue done well ynough: but you thinke you haue not the fraction that you looked for, that is 4/6, yet haue you one equall to it, as by the parts of a shilling you may prooue.

Sc.

Truth it is, for ech of them will bring forth 8 pence, so that, 8/••••, and 4/6 and 2/7, bée all thrée equall. And nowe I perceiue that because 4/6 was not written in the leaste Tearmes that it mighte be, therefore this Reduction broughte forthe not it, but that other whiche is written in the leaste

tearmes. Now vnderstande I this rule wel. But is there anye other waye to worke this Reduction.

Mayster.

Yes, but firste note this,* 1.44 that if you finde no suche diuisor, to reduce the Fra∣ction till you come to 1, because 1 doeth make no diuision, therefore that Fraction is alrea∣die in his least tearmes, as by 71/100 you may prooue and so of 81/98 and many other like. But now for your better ayde to find the due pro∣portion in least tearmes, with more ease for a yong learner, you shall mediate or take the halfe of the Numerator, and also the deno∣minator as long as you may vpon a line al∣wayes parting them with a dashe of your penne as you worke, which maye easilye be done, if the numbers be euen: as 2, 4, 6, 8, or 10: But if they be odde (though it be but in one of them) then must you ab∣breuiate them by ••, by 5, by 7, &c.

And because Examples doe most instruct, I haue in the Page following, set downe the maner of 2 or 3, whose last number at the ende of the lyne, sheweth the least tearme or valuation of that Fraction.

〈 math 〉〈 math 〉

Abbreuiated first by 5, then by 293.

〈 math 〉〈 math 〉

Sc.

Sir I thanke you much, this is verye easie, and good for a yong learner.

Ma.

So it is, but yet notwithstanding, if you can without that diuision by memorie espie the greatest number that maye diuide exactly both Tearmes of your Fraction pro∣poned, then néede you not to vse that diuisi∣on, as in this fraction 6••/96 I sée that 12 is the greatest number that can diuide them both: and therefore without any work, by memo∣rie onely, I turne that into ⅝, but this abi∣litie in knowledge is gotten by exercise.

Yet one other waye of easie Reduction in this kinde there is, when your Fraction hath

any Cipers in y^e first places of both tearmes, then may you by casting away the ciphers, make a bréefe Reduction, as thus 300/400, here take away the Ciphers, and it wil be ¾, which is the same in value with 300/400.

Scholer.

And so if I haue 400/650 it will be 4/65.

Ma.

You are deceiued, for you take away more ciphers from the Numerator,* 1.45 than you doe take from the denominator, whiche you may not doe.

Sc.

I confesse my fault, whiche came of too much haste, I was more gladder of the rule than wise in vsing it: but now I vnderstād it I trust.

Ma.

Then may I go in hand with the fift or last kinde of Reduction••,* 1.46 which teacheth how to turne any fraction proponed into any other denomination that you list: or into a∣ny parts of common coynes, waightes, or measures, or such like.

For declaration whereof, firste you shall mark whether your fraction be a simple fra∣ction, either else a fraction of sundrie parts, I meane of more tearmes than 2. And if your fraction be a fraction of Fractions, or other∣waies compounde, you must reduce it to one

simple fraction. And then marke well the de∣nomination of that other fraction, into which you would turne this, for by that denomina∣tour you must multiplie the Numeratour of your first fraction, and the totall Producte thereof shall you diuide by the denominator of your first fraction, and that quotient shall be the numeratour to the denominatour pro∣poned: as for example I haue this fractiō ⅕, which I would turne into tenth partes, ther∣fore I multiplie this 10 by 3, that is the Nu∣meratour of my fraction, and there riseth 30, which I diuide by 5, and the Quotient is 6, which must be the Numerator to 10, and so ⅗ will be 6/10.

Sc.

This is easie ynough to doe.

Ma.

Then shall you see an other example of the same fraction that is not so easie: as if I would turne ⅗ into viij. partes, proue you that worke.

Sc.

I must multiplie 8 by 3, and there a∣mounteth 24, which I diuide by 5, and the quotient is 4, then is the new fraction 4/8.

Ma.

And see you nothing doubtfull in this worke?

Sc.

I sée that when 24 was diuided by 5, there remayned 4, whiche I did not passe of,

because ye speake nothing of any remayner, but only of the quotient.

Ma.

By likelihoode you remember what I sayd to you in diuision of whole numbers, that you shoulde not passe of the remayner there, but only note it as a summe that could not be diuided without knowledge of Fracti∣ons. Wherefore now marke this, that in all Diuisions of whole numbers, when there is any remainer, you shall set it ouer a line as a Numeratour, and set the Diuisour for the Denominatour, and that Fraction doeth make the Diuision complete, and is parte of the quotient: as if I woulde diuide 48 by 5, the quotient will be 9 ⅗: so in your former worke when 24 was diuided by 5, the quo∣tient should be 4 ⅘, and so the new 〈 math 〉〈 math 〉 Fraction shoulde bée thus: that is 4/8 of the entire number, and ⅘ of 4/8: which you may prooue be example of some coyne.

Sc.

Then I take a Crowne, whose ⅗ is 3 s. Now if I would prooue whether that 3 s bée 4 and ⅘ of 4/8 I shall haue a combrous worke to doe.

Maister.

In déede for whole pennies your example is troublesome: yet tourning

the Crowne into halfe pennies, it is easie y∣nough.

Sc.

Now will I doe it.

Ma.

First let me tell you an easie way how to finde any number that will easily be diui∣ded into such parts as you desire, which way is this. Set down the parts that you desire, and then by one of them multiplie all the o∣ther, the totall whereof shall containe all the partes proponed, as if I would haue a num∣ber that may be diuided into 4, 5, 6, and 7 partes, by 4 multiplie 5, and there riseth 20: then multiplie 20 by 6, & it will make 120: which multiplied by 7, will yield 840: and so of any other numbers.

Sc.

Then in our former example where is mention but of 5 partes and 8 partes, I shall only multiplie 5 by 8, which maketh 40, and that number will serue.

Ma.

So will it.

Sc.

Then what is ⅕ of 40?

Mai.

Proue by the same rule whiche you confesse easie ynough: 3 times 40, is 120, which being diuided by 5, maketh 〈 math 〉〈 math 〉 24 and that is iust.

Now to know whether it be equall to 24, firste I sée by the same rule, that 4/8 is 20, and

2/8 is 5, of which 5, I must take ⅘: 〈 math 〉〈 math 〉 and that by the same Rule is 4. So that I sée now, that is equal to ⅗.

Ma.

And by the waye note this forme of fraction how it is written, that is to say, both the Numerator and his Fraction aboue the lyne, although I know it may be written o∣therwyse, as thus: 4/8 and ⅘ of 4/8, but I ac∣coumpte the other waye more apte a greate deale.

* Scho.

Now I pray you sir let me prooue how this 4/8 and ⅘ of 4/8 of a Crown be equal to ⅘ of a Crown our first proponed fractiō, that hath brought forth these hard fractions to my thinking yet.

First I sée that ⅗ of a crowne is •• s: which is 36 d, or 72 halfe pence. Now if I can find that this fraction 4/8 and ⅘ of 4/8 be equall vnto 3 s: then am I fully answered.

Because I cannot take 4/8 of a Crowne, I turne the crown into half pence, as you wild me, which makes 120, which I diuide by 8, my quotiēt is 15, which takē 4 times makes 60: now resteth me to haue ••/5; of the ⅘, wher∣of ⅛ is 15, that 15 being parted into 5 parts, y^e quotiēt is 3, which takē 4 times makes 12,

which with my 60 before, amounteth to 72: which are then equall to ⅕ my desire.

Ma.

And so may you expresse by an other way then is before mentioned, all 〈 math 〉〈 math 〉 Fractions of Fractions, as thus: That is ¾ of ⅝, and so of other, but I remit these fourmes to the arbitrement of euery wise artes man, to vse as he thinketh most apt and readie.

But now one example more for this rule, and then shall we ende it. If I haue ••/15 of a Soueraine (accounting the Soueraigne 20 shillinges) how many shillinges is that 7/15?

Sc.

I must multiplie 7 by 20, and that maketh 140, which I shall diuide by 15, and the Quotient will be 9 5/15: or else in lesser tearmes, ½.

Ma.

That is 9 s, and one third parte of a shilling, that is 4 d, as by this same rule you may prooue. And this for this time shall suf∣fise for Reduction, saue that I must now re∣peate a little touching Multiplication and Diuision, and so go forward.

MVLTIPLICATION.

IN Multiplication it happeneth sometyme that there be whole numbers to be multi∣plied with Fractiōs: And maye be in two sortes, for eyther the whole number is se∣uerall from the Fraction, and is the Mul∣tiplyer, or else, the whole number is ioyned with one, or both of the Fractions,* 1.47 and so maketh a mixt number thereof. If it be in the first sorte, then néedeth there no Redu∣ction, but onelye multiplie the Numera∣tor of the Fraction by that whole number, and the totall thereof set for the new Nume∣rator.

Scho.

I vnderstand you thus. If I haue 6/2••, to be multiplied by 16, then must I mul∣tiplie that 16 with 6, which is the Numera∣tor, whereof commeth 96, and that must I set for the new Numerator, kéeping still 23 for the Denominator, and so the Fraction

will be 96/2•• that is 4 4/23.

Ma.

And in this sorte of worke you maye abbridge the labour, thus. If it happen the denominator to be such a number, as maye euenly be diuided by the sayd whole number proposed, then diuide it thereby, and sette the quotient of that Diuision for the former de∣nominator: but reserue still the Nume∣rator, and so is the Multiplication en∣ded.

Scho.

Then I fayne this example, 7/20 to be multiplied by 5. And because 5 will iustly diuide 20, therefore I take the quotient of that diuision whiche is 4, and sette in steade of 20, and so the Fraction will be 7/4, that is 1 ¼.

Mayster.

Whiche is all one with 35/••0, that woulde haue followed of the other sort of worke.

Sch.

I perceiue it very well.

Maister.

* 1.48Nowe then for the other sorte where the number is mixt, take this waye: firste to reduce the sayd whole number, and fraction into one fraction Improper (as I shewed you in Reduction) and then multi∣plie them togither, as if they were proper fractions.

Sc.

13 ••/5; being set to be multiplied by ••/9, first I muste reduce the mixte number, as appea∣reth in the margent, by multiplying 13 by 5,* 1.49 and that maketh 65, wherto I must adde the Numerator 3, and so the fraction will be 68/5, whiche nowe I shal multiplie after the accu∣stomed forme, and it wil be 340/40.

Ma

You haue done well: and so maye you sée, that although moste part of the formes of Multiplication maye be wroughte without Reduction, yet some can not, as namelye mixed numbers.

And yet one note more will I tell you of Multiplication, before we leaue it: That is▪ when so euer you woulde multiplye anye Fraction by 2, whiche commonlie is called Duplation, you maye doe it not on∣lie by doubling the Numeratour, but also by parting the Denominator into halfe, if he be euen.

Scholer.

Then if I woulde double 5/••••, I may choose whether I wil make it, 1••/1••, or else ⅚. And in déede I see that all is one, but that the diuiding of the Denominatour seemeth the better waye to make smaller tearmes of the Fraction, and so they shall neede the lesse Reduction.

M.

It is so: and nowe I shall not néede to tell you that Multiplication is proued by Diuision, and Diuision likewayes by multi∣plication, but the like workes that I shewed you in Multiplication, will I shewe you in Diuision also.

DIVISION.

* 1.50WHen any whole num∣ber shall be diuided by a Fraction, you must multiplie y^e said whole number wyth the Denominator of the fraction, and sette the totall thereof for the newe Numerator, and for the Denomi∣nator, let the Numerator of the Fraction.

Scholer.

Then 20 diuided by 〈 math 〉〈 math 〉 wil make 60/3, as here appeareth:

Mayster.

* 1.51Euen so. But if you woulde diuide the fraction by the whole number, thē multiplie the Denominatour by the same

whole number, and sette the totall for the Denominatour, without chaunging the Nu∣merator.

Scholer.

Then to diuide 20/23 by 4, it wyll by 20/92. As here appeareth in this 〈 math 〉〈 math 〉 example.

M.

You say well. And by the same exam∣ple you giue me occasion to remember an o∣ther briefe way to do the same:* 1.52 for if you had diuided the sayde Numerator by 4, and sette the quotient for the Numerator, kéeping stil the olde Denominator, it woulde haue bene not only as well done, but also in a fraction of lesser termes.

Scholer.

I gesse it to be euen so, by a lyke worke that you taughte me in Mul∣tiplication. And for proofe thereof 20/23 being the diuidend, and 4 the Deuisour, I diuide the Numeratour 20 by 4, and the quotient is 5, whiche I sette for 20 ouer 23, thus 5/23. And I sée that it is all one with 20/62, as by diuiding or abbreuing both these tearmes by 4, and so reducing them to their leaste De∣nomination, I maye easi∣lie 〈 math 〉〈 math 〉 proue: as appeareth in this example:

M.

You cōceiue it wel. And if there bée mixt numbers (either one or both) you must firste reduce that mixte number into an improper Fraction. And then worke as you haue lear∣ned.

S.

That was sufficientlie taught in Multi∣plication. Therfore I pray you go forward to some other thing.

Ma.

Then take this note yet for diuision. If the Denominatours be like, then diuide the Numeratours as if it were in whole nū∣bers, and the quotient whether it be Fractiō, whole number, or mixt, is a good quotient for that Diuision. And generallie if one of the Numerators may iustlie diuide the other, by that quotient multiplie the Denominator of the lesser Numerator, and set it that doeth a∣mounte in the roome of y^e same denominator, and then for a Numerator to it, set the deno∣minator of the other fraction.

Scho.

Then if I woulde diuide ••/4 by 12/1•• I sée that 3 will diuide 12, and the quotient wil be 4, by whiche I must multiplie the o∣ther 4 that is the Denominator vnder 3, and then it is 16, whiche I set for the Denomi∣nator 4, and ouer it in steade of the ••, I must set 17, the other Denominator, and so is it

thus, 17/16.

Maister.

And so is 17/16 in steade of 11/48, which woulde haue risen by the com∣mon 〈 math 〉〈 math 〉 worke: as here appea∣reth:

And now for Mediation (which is to diuide by 2) marke this: If the Numeratour be e∣uen, set the halfe of it in his place without the Diuisour, and so haue you done: and if the Numeratour be not euen, then double the de∣nominator.

Sch.

That is if I would mediate 6/11 I may make the quotient ••/11. And if I woulde medi∣ate ••/11 I must make it 7/22.

Mayster.

Nowe truste I that you haue sufficiente knowledge in Reduction, Mul∣tiplication, and Diuision: and therefore will I goe in hande wyth Addition and Sub∣traction, whyche nowe will appeare easie ynough.

ADDITION.

* 1.53WHensoeuer you haue any fractions to be ad∣ded, you must consider whether they be of one Denomination or not▪ And if they be of one denomination, then adde the Nume∣rators togither, and set y^e that amounteth, for the Numerator ouer the common Denomi∣nator, and so you haue done. The reason is, bicause that suche differ little in Addition or Subtraction from the worke of vulgare de∣nominations, where the Denominators bée no numbers: as 3 pence and 5 pence, make 8 pence,* 1.54 where y^e denomination is not altered. But & if the fractiōs be not of one denomina∣tion, or any of them be mixt of whole nūbers and fractions, then must you first reduce them to one denomination, and after adde them. And if they be manye, then adde first two of them, and to the sum that doth amount of the Addition, add the third, and then the fourth, & so forth, if you haue so many.

Sc.

This séemeth easie ynough, now that I haue alreadie learned to multiplie and to re∣duce, without which two, I could neuer haue wrought this. And therefore now I see good reason, why you did place Multiplication and Reduction before Addition.

Maister.

It is wel considered, but yet refuse not to expresse your vnderstanding of it, by an example.

Sch.

Then would I adde first 7/18 with 5/18, and because the Denominators are like (and so néedeth no Reduction) I adde 7 to 5, which maketh 12, and then is my sum 12/18, that is in smaller numbers ⅔.

And if I haue many numbers to be added, as here 34/85 9/10, firste I must reduce them (by∣cause they haue diuerse denominatours) into one denomination, and then will they bée thus. 〈 math 〉〈 math 〉 or in lesse tearmes 15/4•• 32/40 26/40, whiche by Addi∣tion do make 83/•••• that is 2 3/••••.

M

Now may we go to subtraction.

SVBTRACTION.

SVbtraction hathe the same precepts that ad∣dition hadde, for if the Denominatours bée like, then muste you subtracte the one nu∣meratour from the o∣ther, and the rest is to be set ouer the common denominator, and so your Subtraction is ended: but and if you haue manye Fractions to be subtracted out of many, then must you reduce them to one de∣nomination, and into two seueral Fractions, that is, all that muste be subtracted into one Fraction, and the residue into an other Fraction, and then worke as I sayde bée∣fore.

Scholer

For the first example I take 15/12 to be subtracted out of 17/12, and the reste will bée ••2/•••• or ⅙.

For an other example I take 2/4 to be sub∣tracted out of 7/3 which I must reduce, and it wil be thus, 24/••2 and ••8/••2.

Then doe I subtracte 24 out of 28, and

there resteth 4, which I set ouer the common Denominator for a Remayner, thus, 4/32, that is ⅛.

Now for the thirde example, I take ¾ and ⅚ to be subtracted from ⅞ and 9/10. And bycause their denominators be diuers, I do reduce thē thus, 1••••••/1920▪ 16••••/1720, ••••80/1920 17••8/1920.

Then do I adde the two first, & they make 3••••••/19••••. Also I adde the two laste, and they yéeld ••••08/162•• Then doe I subtract 3040 out of 3408, and there resteth 368, so is the remainer ••68/1920 that is in smaller Tearmes, 23/120. And thus haue I done with Subtraction, excepte you haue any more to teach me.

Mai.

Proue one example more of two Fra∣ctions of diuers denominations.

Scho

I take these two Fractions, ⅞ and 7/24, which being reduced, wil stād thus, 168/192. & 72/191 Now would I subtract 168 out of 72, but I can not.

Ma.

Then maye you perceiue that you mistooke the Fractions: for you cā neuer sub∣tract the greater out of the lesser, althoughe you may adde, multiplie or diuide the greater with the lesser. And albeit that ⅞ hathe bothe hys tearmes lesser than 9/24, yet is 9/2•• the les∣ser Fraction: for generallie if you multiplie y^e

Numerators and denominators of two fra∣ctions crosse ways,* 1.55 that Fraction is the grea∣test, of whose numerator commeth the grea∣test summe, as in this example: 7 multiplied by 24, maketh 168: and 9 being multiplyed by 8, yeldeth but 72 therefore is the first Fra∣ction ⅞ the greatest of these two, so can you not subtract it out of a lesser Fraction.

But and you should subtract a fraction out of a whole number what would you do?

Sc

Marrie I woulde reduce the whole nū∣ber into a Fraction of the same Denomina∣tion that my Fraction is, and then worke by Subtraction.

M.

So may you doe, but it is easier much, if your fraction be a proper fraction, that is to say, lesse then an Vnite, to take an Vnite frō the whole number, and then turne it into an Improper Fraction▪ and so worke your sub∣traction. As if I woulde subtracte ⅗ from 4, I maye take one from 4, and turne it into 5/5, from whiche if I bate 7/6, there will remayne 3 ⅖: And if the firste fraction be an Improper Fraction, then may I take so manye Vnites from the whole number, that they may make an improper fraction greater then that firste, and then worke by Subtraction: As if there

be proponed 10/3 to be subtracted frō 6, because ••0/3 is more than 3, and not so much as 4. I muste take 4 from 6, and turne them into thirdes thus. 12/3 then abate 10/3, and there re∣steth ⅔ so the whole remainer is 2 ⅔. Or, else you may at your pleasure take 3 ⅓ whiche is 10/3 from 6 whole: Then set one vnder 6. as thus 6/1: And then to reduce those 2 fractions into one Denomination as 〈 math 〉〈 math 〉 here appeareth:

Then 10/3 from 18/3 resteth 8/3: which maketh 2 ⅔ youre desire:

And thus wil I make an end of the works of commō Fractions for this time, not doub∣ting, but you can applye them both vnto the rules of Progression, and also vnto the Gol∣den rule, without any other teaching thē you haue learned before, which might séeme tedi∣ous to repeate, saue that in some special di∣uersities, which be peculiar to Fractions, I can not ouerpasse, but instruct you somewhat by the way.

THE GOLDEN Rule.

THerefore as touching the Golden rule for the placing of the 3 numbers proponed in the question, wherby to finde the thirde, and for the forme of their worke, with other like notes, I refer you to that which you haue alreadie learned.

But this easie forme of working by fracti∣ons shal you note, y^t if your thrée numbers bée fractions, for an apt worke and certain, mul∣tiply the Numerator of the first nūber in the question, by the Denominator of the seconde: And all that againe multiplie by the denomi∣nator of the third number, and the total ther∣of shall you kéepe for to be the Diuisor. Then multiplie the Denominator of the first num∣ber by the Numeratour of the second, and the whole therof by the Numerator of the third, and the totall thereof shall be youre diuidend. Now diuide this diuidend by y^e diuisor which

you founde out before, and that number shall be the fourth number of the question whiche you séeke for: As in this exāple. If ¾ of a yard of Veluet cost ⅔ of a Soueraigne (esteemed at 20 shillings) what shal ⅚ cost?

Scholer.

If it please you to let me make the answere,* 1.56 I woulde first place 〈 math 〉〈 math 〉 these thrée numbers, as I learned in whole numbers thus.

And then according to your newe rule, I must multiplie 3 being Numeratour in the first number, by thrée the Denominatour of the seconde, and thereof commeth 9, which I multiplie againe by 6, the Denominatour of the third number, and so haue I 54. which I kéepe for the Diuisor, then multiplie I 4, the Denominator of the first, by 2, the Numera∣tour of the seconde, and there riseth 8, whiche againe I multiplie by 5, the Numeratour of the thirde, and it maketh 40: then must I di∣uide 40, by 54, and it wil be 40/••4 that is 20/27, in lesser tearmes, and then the figure wil stand thus.

But what that is in 〈 math 〉〈 math 〉 money, I can not tell except I shal worke it by Reduction, as you taught me.

Maister.

It forceth not nowe, you may re∣duce it when you list, but it were disorderlie done here to mingle diuerse workes togither, where we do not séeke the value of the thing in common money, but in an apt nūber, whi∣che you haue wel done. And therefore will I yet shew you an other like way of easinesse in worke, how you may chaunge your 3 Fra∣ctions into 3 whole numbers, by whyche you shall worke as if the question were proponed in whole numbers. The first nūber you shall finde as I taught you: now to finde the diui∣sor of the seconde number, take the Numera∣tor for the secōd fraction: and for the third nū∣ber take that, that riseth of Multiplication of the Denominatour of the firste, by the Nu∣merator of the thirde, and then worke youre question.

Scho.

* 1.57For example hereof, I put this que∣stion, If 11/12 of 1 lb waighte of siluer, be worth 11/4•• of a Soueraigne what is ½ of 1 lb. weighte worth? For the aunswere, 〈 math 〉〈 math 〉 first I place the Fractions in order thus.

Then to turne these fra∣ctions into whole numbers, I multiplie 11 whiche is the Numerator of the first, by 4 (y^t

denominator of the seconde) and there cōmeth 44, whiche I multiplie by 2 the Denomina∣tor of the third, and so amounteth 88, whiche I set for the Diuisor in the first place. Then in the seconde place I set 12, whiche is Nu∣merator in the seconde Fraction, and in the third place I set the summe that amounteth of 12, being the Denominator in the firste number, multiplyed by 1, being numeratour in the third number, & so the figure 〈 math 〉〈 math 〉 wil stand as here you sée.

Then to worke it forth, I multiplie 12 by 12, and there amounteth 144, which I diuide by 88, and the quotient wil be 1 ••6/88, or in les∣ser termes, 1 7/11 and then the 〈 math 〉〈 math 〉 figure wil stand thus.

Ma.

These ij. formes nowe you vnderstand wel ynough And as for any other, at this time I will not repeate, onlie this shall you marke for y^e profe of this rule, whether your worke be well wrought or no. Multiplie the first number by the fourthe, and note what amounteth:* 1.58 then multiplie the seconde by the third, and marke what amounteth also Now if those two nū∣bers so amounting be equall, then is youre work wel done, else you haue erred. And this

shal suffise for the former rule,* 1.59 but in the bar∣ker rule, this shal you note for ease of worke, that you multiplie the Numerator of the first by the Numerator of the second, and y^e whole thereof by the Denominator of the third, and that amounteth therof, shal be the Diuidend. Then multiplie the Denominator of the first by the Denominator of the seconde, and that whole by the Numerator of the third, & that riseth therof shall be the diuisor. Example of this: I did lende my friend ¼ of a Porteguise vij. monthes, vpon promise that he should doe as much for me againe:* 1.60 and when I shoulde borow of him he coulde lende me but 5/12 of a Porteguise, now I demaund how long time must I kéepe his money in iuste recompence of my lone, accompting 13 monethes in the yeare?

Scholer

The first number must be the first money borowed, that is ••/4 of the Porteguise: the second number the 7 monethes, that is 7/••3 of a yeare: and the thirde number the money that was lent in recompence, that is 5/12 of a Porteguise: then I set the numbers 〈 math 〉〈 math 〉 thus.

Then (as you taught me) I mul∣tiplie thrée (being Numeratour in the firsts

number) by 7 the Numerator of the seconde number, and it maketh 21, which I multi∣plie by 12, the denominator of the 3 & so haue I 252 for the diuidēt: thē I multiply 4 y^e de∣nominator of the first, by 13 the denomina∣tor of the second, and it yéeldeth 52. whiche I multiplie agayne by 5, the Numeratour of the third, and it will make 260, that is the Diuisor. Then must I diuide 252, by 260, so it wil be in y^e smallest fractiō, 63/65 of a yeare.

Ma.

And this doe you sée some ease in wor∣king, better than to multiplie and diuide te∣diously so many Fractions. An other questi∣on yet will I propose, to the intent you may sée thereby the reason of the statute of assise of bread and ale, which in al Statute bookes in Frenche, Latine and Englishe,* 1.61 is muche corrupted for wāt of knowledge in this art: for the right vnderstanding whereof I pro∣pone this question.

When the price of a quarter of Wheate is 2 s the farthing white lofe shal wey 68 s.* 1.62 then I demaunde, what shall such a Loafe weye, when a quarter of Wheate is solde for 3 s?

Scho.

This Question must be wroughte as it is proponed in whole numbers and not

in Fractions.

Mai.

You séeme to say reasonably, how be it, in that Statute of Assise, the rate is made by the proportion of parts in a pound weight Troie, else could it not be a Statute of anye long continuaunce, séeing the shillinges doe chaunge often, as all other monies doe: but this Statute being well vnderstanded, is a continuall rule for euer, as I will anon de∣clare by a new table of Assise, conuerting the shillinges into vnces and parts of vnces. Therfore here by a shilling you must vnder∣stand 1/20 of a pound weight, and so by pence 1/2•• of an vnce, wherefore although ye mighte worke this question proponed by whole nū∣ber well ynough, for y^e time when the statute was made, yet to applie it to our time, and to make it to serue for all tymes generallye it is best to worke it by fractiōs, setting for 2 shilling 2/20: and for 68 shillings, 68/20: and so for thrée shillings 3/20, and then will the fi∣gure of the question stand thus. 〈 math 〉〈 math 〉 In which question because all the denominators be like, you shal worke onely with the numerators

Sc.

Then I shall multiplie 68 by 2, wher∣of commeth 136, which if I diuide by 3, the

quotient will be 45 ⅓: but how shal I make a Fraction of that to stand with the other?

Ma.

Haue you so soone forgottē what was taught you so lately? This is his forme.

Sc.

I remember it now and then 〈 math 〉〈 math 〉 it signifieth 45 twentie parts, and the third deale of one twentie part.

Ma.

So is it, and that maketh in shillings, 45 s 4 d: wherby you may note one greate error in the Statute bookes, which haue con∣stantly 48 s in that Assise. And by this rule, if you examine the Statute you shall finde many summes false, wherefore for the true vnderstanding of that statute and suche lyke as I haue made mention of it, and somwhat recognised it, so doe I wish that al gentlemē and other studentes of the lawes, would not neglect this arte of Arithmetike as vnnéede∣full to their studies. Wherfore to encourage them thereto, and to gratifie both them and all other in generall, I will exhibite a Ta∣ble of that part of y^e statute in two columnes, and in a third columne I wil ad the correcti∣on of those errors which haue crept into it.

Here followeth the Table.

The price of a quarter of vvheate.	The vveight of a far∣thing vvhite lofe by the statute bookes.	The correcti∣on by iust Assise.
s. d.	li. s. d.	li s. d.
1 0	6 16 0	6 16
1 6	4 10 8	4 10 8
2 0	3 8 0	3 8 0
2 6	2 14 4 1/••	2 14 4 ⅘
3 0	2 8 0	2 5 4
3 6	2 2 0	1 18 10 2/••
4 0	1 16 0	1 14 0
4 6	1 10 0	1 0 2 2/••
5 0	1 8 2 ½	1 7 2 2/••
5 6	1 4 8 ¼	1 4 8 ••/••••
6 0	1 2 8	1 2 8
6 6	0 19 11	1 0 11 1/13
7 0	0 19 1	0 29 5 1/••
7 6	0 18 1 ½	0 18 1 1/••
8 0	0 17 0	0 17 0
8 6	0 16 0	0 16 0
9 0	0 15 0 ¼	0 15 1 ⅓
9 6	0 14 4 ¼	0 14 3 15/19
10 0	0 13 7 ½	0 13 7 1/••
10 6	0 12 11 ¼	0 12 11 3/7
21 0	0 12 4 ¼	0 12 4 4/••••
11 6	0 11 10	0 11 9 2••/2••
12 0	0 11 4	0 11 4

In the common bokes there is no farther rate of assise made, than vnto 1 2 s the quarter of wheate: but in an aunciēt copie of 200 years olde (which I haue) there is added the rate of assise vnto 20 s the quarter, but yet was y^t assise also eyther wrong cast at y^e first pēning, or els corrupt sith that time, for lacke of iuste knowledge in the rule of proportion, whiche I wil adde here also, to gratifye suche as be desyrous to vnderstande truth exactlye.

The price of a quarter of vvheate.	The vveight of the far∣thing vvhite lofe by the statute bokes.	The core∣ction of the errors.
s. d.	s. d.	s. d.
12 6	11 0	10 10 14/25
13 0	15 0 ½	10 5 7/1••
13 6	10 1 ½	10 0 8/••
14 0	9 7	9 8 4/••
14 6	9 2 ½	9 4 16/20
15 0	9 1 ½	0 0 ⅘
15 6	9 1 1/••	8 9 9/3••
16 0	9 0	8 6 0
16 6	8 6	8 2 10/11
17 0	8 3	8 0 0
17 6	7 10	7 9 9/••5
18 0	7 6	7 6 ⅔
18 6	7 3	7 4 8/••7
19 0	7 2	7 1 17/19
19 6	5 10	6 11 9/1••
20 0	5 6	6 9 ⅗

These 2 tables I haue set seuerall, because no man should think that I would either ad or take away frō any law those parts which might of right séem either superfluous eyther diminute, but yet I may not be so curious as to neglect manifest errors, which is not only my part, but euery good Subiects dutie with sobrietie to correct. And for auoiding of offēce I haue rather don it in this priuate booke ra∣ther than in any booke of y^e statutes self, tru∣sting that all men will take it in good parte.

Sc.

I would wish so, but I dare not hope so sith neuer good man y^t would reforme error, could escape y^e venemous tōgs of enuious de∣tractors, which because they either cannot or list not to do any good thēselues, do delight to bark at y^t doinges of other, but I beséeke you to stay nothing for their peruerse behauiour.

Ma.

I cōsider many things y^t some may ob∣iect, wherevnto I am not vnprouided of iust answers, but I wil not séem so hasty to make y^e answers before I heare their obiectiōs, but as I trust that men are of a bettter nature, & more grateful now thā som hath bin in time passed, as I haue don in y^e statute of Assise for bread in rate of s, so wil I set forth the like table in pounds & ounces, & the parts thereof,

that it may be easilie applyed to all tymes: but I mean not by this to alter any word of the statute (being so good an ordināce, & of so great continuance) but onely to make it as a kind of expositiō & declaratiō of y^e said statut, trusting y^t therby y^e statute may be better vn∣derstād, & consequētly better put in executiō. And here you shall note,* 1.63 y^t I haue accounted the shillings after the rate of lx s, to the poūd weight, bicause I estéem it y^e most apt rate for our time. Wherfore if in y^e first column you find y^e price of wheat, directly against it in y^e second column, you may find y^e weight of the farthing white lofe, in this our time: & if you double y^t number (as I haue done in y^e thirde columne) then haue you the weight of y^e half pēny white lofe & so in y^e fourth column is set y^e weight of y^e pēny white lofe. It néedeth not to tel you that, y^t the sight doth testifie, how y^e euery colūn is parted into 3 smaller pillers, wherof y^e first colūne hath these 3 titles, poū∣des, shillings, & pennies: y^e other 3 columnes haue ech of thē these 3 titles, pounds, vnces, & pēnie weights. And as in the first colūne xij d make a s, & 20 s make a pounde, so in the other iii columes xx pence weighte maketh an vnce. and xij ounces doe make a pounde.

The price of a quarter of wheate.

lb	s	d.
0	3	0
0	4	6
0	6	0
0	7	6
0	9	0
0	10	6
0	12	0
0	13	6
0	15	0
0	16	6
0	18	0
0	19	6
1	1	0
1	2	6
1	4	0
1	5	6
1	7	0
1	8	6
1	10	0
1	11	6
1	13	0
1	14	6
1	16	0
1	17	6
1	1••	0

The weight of the farthing white loafe.

lb.	vnc.	d w.
6	9 ½	2
4	6 ¼	3
3	4 ¾	1
2	8 ½	2 ⅘
2	3	4
1	11 ¼	1 2/7
1	8 ¼	3
1	6	2 ⅔
1	4 ¼	1 ⅖
1	2 ¼	1 8/11
1	1 ½	2
1	0 ½	1 1/11
0	11 ½	3 1/7
0	10 ¾	2 ⅕
0	10	4
0	9 ½	2
0	9	1 ⅓
0	8 ½	1 15/1••
0	8	3 ⅕
0	7 ¼	0 3 〈 math 〉〈 math 〉
0	7 ¼
0	7	1 21/21
0	6 2/4	1
0	6 ½	0 14/15
0	6 ½	0 ••/••

The weight of the halfe pennie white loafe.

ll.	vnc.	d. w.
1 3	7	4
9	0 ¾	1
6	9 ½	2
5	5 1/••	0 ⅗
4	6 ¼	3
3	10 ½	2 4/7
3	4 ¾	1
3	0 1/••	0 ⅓
2	8 ½	2 ⅘
2	5 ½	3 5/11
2	3	4
2	1	2 2/13;
1	11 ¼	1 2/7
1	9 ¾	0 ⅕
1	8 ¼	3
1	7	4
1	6	2 ⅔
1	5	3 11/19
1	4 ¼	1 ⅖
1	3 ½	0 6/7
1	2 ¾	1 ••/11
1	2	3 19/29
1	1 ½	2
1	1	1 1/25
1	0	1 1/12
1	0	1

The weight of the penny white lofe.

li.	vnc.	d. w.
27	¼	3
18	1 ½	2
13	7	4
10	10 ½	1 ⅕
9	0 ¼	1
7	9 ¼	0 1/7
6	9 ½	2
6	0 ½	0 ⅔
5	5 1/••	0 3/5
4	11 ¼	1 10/••1
4	6 ¼	3
4	2	4 4/13
3	10 ½	2 4/7
3	7 ½	0 ⅗
3	4 ••/4	1
3	2 ¼	3
3	0 ¼	0 1/3
2	10 ¼	2 ••/19
2	8 ½	2 ⅘
2	7	1 ••/7
2	5 ½	3 5/11
2	4 ¼	2 15/23
2	3	4
2	2	2 6/25
2	1	2 2/13
2	••	••

Sch.

Sir, I doe thanke you most hartily for this, not only in mine owne name and in the name of all Studentes, but also in the name of the whole Commons, to whom the restitution of this Assise (I trust) shall bring restitution of the weight in breade, whiche long time hath bene abused. And if you know any like thinges more, wherein you woulde vouchsafe to declare the errors and set forth the truth, you cannot but obtain great thāks of all good harted men that loue the common wealth.

Mai.

I haue sundrie thinges to declare but I haue reserued them for a priuate booke by it selfe, yet notwithstanding because the sta∣tute of the rate of measuring of ground is so common that it toucheth all men, and yet no more common than néedefull, but so muche corrupt, that it is to farre out of all good rate, not only in the Englishe bookes of Statutes commonlye printed, but also in the Latine bookes, and in the Frenche also, for I haue read of eche sorte, and conferred them diligēt∣lye, I will giue you a Table for the restitu∣tion of those errours, as maye suffise for this present time. And first will I propose one question to you touching the vse of that Sta∣tute,

whereby you maye perceiue the order how to examine the whole Statute,* 1.64 and eue∣ry parcell thereof, and the question is this.

When the Acre of grounde doth contayne foure perches in breadth, then must it con∣tayne 40 perches in length: then doe I de∣maunde of you, howe much shall the length of an Acre be, when there is in the breadth of it 13 perches? but before you shall aun∣swere to this question, I will declare vnto you an other Statute, whiche is the grounde of the former Statute. And that Statute is this. It is ordayned that 3 Barlye Cornes, drye and rounde, shall make vp the measure of an inche: 12 ynches shal make a foote,* 1.65 and 3 foote shall make a yarde (the common En∣glish bookes haue an elne) fiue yardes and a halfe shall make a perche, and 40 perches in length, and 4 in breadth, shall make an Acre. This is that statute: whereby you may per∣ceiue that the intent of the Statute is,* 1.66 that one Acre shoulde containe 160 square Per∣ches. Now let me heare you aunswere to the question.

Scholer.

As I perceiue by the wordes of that Statute, a perche to be 1/160 of an Acre, so could I make those numbers all in Fra∣tions,

and so worke the question: but séeing I may doe it also in whole numbers I take that forme for the moste easie, 〈 math 〉〈 math 〉 therefore thus I set the questi∣on in forme. Then doe I mul∣tiplie 40 by 4, and it maketh 160, which I diuide by 13, and the quotient is 12 4/1••.

Ma.

Now turne that 4/1•• into the common partes of a perche, as they be named in the former Statute: how be it, it shall be best to take one of the least partes in Denominati∣on for auoyding of much labour, as féete, whereof the perche contayneth 16 4/••.

Sc.

Then to turne 4/13; into feete, I Multi∣plye 16 ½ by 4, and it maketh 66, whiche I muste diuide by 13, and the Quotiente is 5 1/13.

Mayst.

So I finde that if the acre holde in breadth xiij perches it shall containe in lēgth 12 perches, 5 foote, and 1/1•• of a foote, which is not fully an ynche, for the ynche is 1/12 of a foote.* 1.67 But here all the Statute Bookes in Latine and Englishe (that I haue séene) doe note it to be 13 perches, 5 foote and 1 inche: which maketh aboue 13 perches to many in the acre, so that I woulde haue thoughte the

errour to haue crept into the printed bookes by the great negligēce that Printers in our time doe vse, saue that in written Copies of great antiquitie, I doe find the same. Yet haue I one frenche Copie, which hath 12 per∣ches ••/••, and one foote, and that misseth verye little of the truth.

Sc.

Then I sée it is true that I haue oftē heard say, that the truest copies of y^e statutes be the Frenche copies.

Ma.

That is often true, but not generally, as I haue by conference tried diuerslye: but in this statute the frenche booke is most cor∣rupt in all other places lightlye.

But now to performe my promise I will set forth the Table for measuring of an Acre of ground onely by suche partes as the Sta∣tute doth mention, because at this time I doe of purpose write it for the better vnderstan∣ding of the statute, and hereafter with other thinges I intende to set forth this same more at large.

In this Table following, I haue not done as in the other statute before cōpared by re∣stitution with the faults crept into the Sta∣tute, but onely haue written that true mea∣sure, which the equitie of the Statute doeth

pretends. For it were to vile to iudge of so noble Princes and worthie Councellors, as haue authorised & set forth this statute, that they would make one acre in any form grea∣ter than an other, but euery one to be iust and equall with ech other, which is the grounde also of my worke, and hereby maye all men perceiue how néedefull Arithmetick is vnto the Studentes of the law. But now I think best to make an end of these matters for this present time, sith the table hath in it none ob∣scuritie, that I should néede to declare.

The breadth	The length of the acre.
perde.	perde.	feete.
10	16	0
11	14	9
12	13	5 ½
13	12	5 1/13
14	11	7 1/14
15	10	11
16	10	0
17	9	6 27/34
18	8	14 ⅔
19	8	16 18/19
20	8	0
21	7	10 3/14
22	7	4 ½
23	6	19 18/23
24	6	11
25	6	6 5/••
26	6	2 7/13
27	5	15 5/18
28	5	11 11/14
29	5	8 31/58
30	5	5 ½
31	5	2 41/62
32	5	0
33	4	14
34	4	11 11/17
35	4	9
36	4	7 ⅓
37	4	5 13/37
38	4	3 9/19
39	4	9/13
40	4	0
41	3	1 63/82
42	3	13 6/42
43	3	11 77/86
44	3	10 ½
45	3	9 ••/••

Sc.

In deede Sir, I vnderstand the Table (as I thinke) by those other whiche you sette forth before. For in the first Columne is set the perches of the breadth of any Acre, and then in the 2 columnes following appeareth how many perches and how many foote the same Acre must haue for his length

Ma.

You take it well: how be it to speake exactly of breadth & length, the first columne doth sometime betoken the breadth: & some∣time the length,* 1.68 for properly the longest side of any square doth limit his length, and the shorter side doth betokē the breadth, yet it is no great abuse in such tables, where a man cannot well change the title, to let the name remayne, although the proportions of the numbers doe chaunge: for still by the first co∣lumne, is expressed the measure of the one side, and by the two other pillers in one Co∣lumne, is set forth the measure of the other side. And this shall be sufficient now for the vse of the golden Rule.

NOwe somewhat will I touch certaine other rules, which for theyr seuerall names maye séeme diuerse rules & distinct from this, but in déede they are but braunches of it: yet bi∣cause they haue not onelie seueral workings in appearance, but also pleasant in vse, I will giue you a taste of eche of them. As for the rule of Felowshippe, both single and double, with time and without time, I shall néede to saye little more than I haue alreadie sayde in teaching the workes of whole numbers, yet an example or .ij. wil we haue to refresh the remembraunce of the same, and to declare certaine proper vses and applications of it, as this for one.

Foure men get a bootie or prise in time of warre, the prise is in value of money 8190 lb,* 1.69 and bicause y^e men be not of like degree, ther∣fore their shares may not be equall, but the chiefest person wil haue of the bootie the third part, and the tenth parte ouer: the second will haue a quarter and the tenth part ouer: y^e third

will haue the sixt part: and so there is lefte for the fourth man a verie smal portion but such is his lot, (whether he be pleased or wrothe) he must be content with one xx. parte of the praye. Now I demaunde of you, what shall euerie man haue to his share?

Sch.

You must be faine to answere to your owne question, else is it not like to be aun∣swered at this time.

Maister.

The forme to vnderstande the so∣lution of this question, and all suche like, is this:* 1.70 Reduce al the Denominators into one number by Multiplication, except that anye of them be partes of some other of them, for al suche partes you may ouerpasse, and take for them all those numbers, whose parts they be: as in this example the shares be these ⅓ 1/10 ¼ 1/12 ⅙ 1/20 if I multiplie al the Denominatours togither, beginning with 3, and so go on vn∣to 20, it will make 144000: but considering that 3 is a part of 6, I shall ommitte that 3, and likewayes 10, whiche is a part of 20, I maye ouerpasse also, and then is there but 3 denominatours to multiplie, that is 4, 6, and 20, whiche make 480, whiche summe I take for my worke, bycause all the Denomina∣tours wil be founde in it. Then I take suche

partes of it as the question importeth, that is for the firste man, ⅛ 1/10, the ⅓ is 160 the 1/10 is 48: whiche I put in one summe for the firste mans share, and it maketh 208. Then for the seconde mans share, I take ¼, whiche is 120, and 1/10. whiche is 48, and that maketh in the whole 168. Now for the thirde man whiche muste haue ⅙ I take 80 And for the fourth man there remayneth but 24, whiche is 1/20 of the whole summe: so that if the whole pray had bene but 480 lb. then were the que∣stion answered: but bicause the summe was of greater value, by this meanes nowe shall I know the partition of it. I must set my nū∣bers by the order of the Golden rule, putting in the firste place the number that I founde by multiplying the Denominatours, and in the seconde place the summe of the bootie.* 1.71 And looke what proportion is betwéene the firste number and that seconde, the same proporti∣on shall be betwéene the partes of that first number and the partes of the seconde, com∣paring eache to his like. Therefore I muste put in the thirde place, one of the partes or shares, and then worke by the former rule of proportion or Golden rule. And bycause I

haue 4 seueral partes of the first number, by which I would finde out 4 like partes of the seconde number, therefore muste I make 4 seueral figures.

Scholer.

Nowe I truste I can aunswere to your question, as by your fauoure I wyll proue.

〈 math 〉〈 math 〉

And to trie it, I set the 4 figures thus, mar∣ked with A, B, C, D, to shewe their order. And then in eache of them I multiplie the seconde number by the thirde, and diuide their totall by the first, and so amounteth the fourth sum whiche I séeke for, for if I do multiply 8190 by 208, it maketh 1703520, whiche being diuided by 480, maketh in the quotient 3549 for the first mans portion: And so workyng with the other thrée figures, I find for the se∣cond man 2866 ½, and for the third mā 1365: and then for the fourth man 409 ½. And so is

euerie mans share set forth in the figure here annexed.

〈 math 〉〈 math 〉

And thus I thinke I haue done wel.

Ma.

If you misdoubte your working and liste to proue it, adde all the shares togither: and if they make the totall, thē séemeth it wel done.

Sc.

I maye sette them thus:* 1.72 〈 math 〉〈 math 〉 and then by Addition the iuste summe doeth amounte, that is 8190, and therefore (as you saye) it séemeth to be wel wrought.

But I beséeche you, is there anye doubte in this tryall, that you vse that word,* 1.73 Sée∣meth?

Maister.

You maye easilie coniecture, that if you did assigne the firste mans share to the laste, and so chaunge all the rest, that

one had an others share, yet woulde the Ad∣dition appeare al one, and therfore is not the proofe exact.

But if you wyl make a iuste proofe for the firste mans parte take ••••/•••• of the whole sum, and if it agrée with the number in the fi∣gure, then it is well done. And so doe for the seconde, thirde, and fourth summes, and this proofe faileth not. Nowe will I propounde certaine other questions whiche haue bene sette foorth by certaine learned men, albeit not without some ouersighte whiche questi∣ons I protest heartilie I doe not repeate to depraue those good men, whose labours and studies I muche prayse and greatelie de∣lighte in, but onelye according to my pro∣fession, to séeke oute trueth in all thinges, and to remoue all occasions of errour, as muche as in me lyeth: and for that cause I will onelie name the questions wythout hurting the Authours name. The firste que∣stion is this.

* 1.74Foure men did builde a house, whiche coste them 3000 crownes, their shares were such, that one man should pay 1/•• of the summe, and 6 crownes ouer: the seconde should pay ⅓ and 12 crownes ouer: the third man must lay out

⅔, abating 8 crownes and the fourth mā shold pay ¼, and 20 crownes more, cā you answere to this question?

Sc

No in good faith sir, and that you knowe best of anye man, for I knowe no more than you haue taught me.

Mayster.

Then I dare say you can not doe it neither yet the best learned man that euer did propose it, for the question is impossible:* 1.75 for declaration whereof I wil be bolde to vse first the representation of the nūbers in their aptest form, (Although I haue not yet taught you that manner of worke) bicause it maye apppeare plainelie that the question is not possible, for here I haue 〈 math 〉〈 math 〉 set the parts, and added them, and they make the whole summe and ¾ and 30 more. Now howe is it possible to diuide true∣ly either gaines eyther charges so, that the particulars shall be more than the total.

Sc.

It is against the forme of proofe by Addi∣tion of partes.

Maister.

You say truth. And bycause you

shal perceiue it the better, I 〈 math 〉〈 math 〉 wil trie it after the vulgare forme, as in this figure you see where the ½ with 6 ouer is 1506: for the totall is as you heard before 3000: the ⅓ and the 12 more, is 1012: the ⅔ woulde be ••0••0, but then abating 8, it is but 1992, and then laste of al, the ¼ is 750, and the 20 more maketh 770: which al being added in one summe, doe make 5280, where the totall summe shoulde be but 3000, which sum if you diuide by 4/3, so shall you haue ¾ of it, that is 2250, and thereto adde 30 more, then wil those 3 summes make 〈 math 〉〈 math 〉 5280: wherby you may see how this forme as well as the other, doeth declare that the particu∣lars in that question woulde make more than the whole sum by ¼, and ••0 more: and therefore can that que∣stion not be accepted as a possible thing, but yet doe certayne learned men propound suche questions, and aunswere to them. Therefore somwhat to say to their excuse rather of their good meaning than for their doing, I will a∣none declare what may be saide for their de∣fence:

but in the meane season I will pro∣pounde the question as it may be wrought by good possibilitie. As if foure men build a house togither, and it cost them 3000 crownes, and then for the partition they agrée thus: that as often as the firste man doth paye 6 crownes, so often the seconde shall paye foure, the third man 8 and the fourth man thrée. Or els thus: that the firste man shall pay double so muche as the fourth, and the seconde man shall paye ⅔ of the firste mans charge: the third manne shall pay double so much as the second: (And these two wayes are to one ende) but further for their agréement it is appointed also, that the firste man shal giue 6 crownes ouerplus, and the seconde 12, and the fourth shal giue 20, but the third man shal giue no ouerplus, but shal haue 8 crownes abated of his charge. Now is the question possible to be assoyled, and this is the way to doe it. Marke the pro∣portion of the seuerall charges,* 1.76 and set out small numbers in that rate, by whiche you may reduce the worke to the Golden rule, as here in the first forme, the nūbers are alredie named, 6, 4, 8, 3: and in the seconde forme, although they be not plainly named, yet they may be the same nūbers: for 6 is double to 3,

and 4 is ⅔ of 6: and againe 8 is double to 4. Now adde these togither and they make 21, whiche 21 must be set in the first number in the Golden rule: for if it with the ouerplus of eache mans charge woulde make the totall sū of the charges, thē were those seueral sums the charges of ech man, besids his ouerplus: but now it is not so.

* 1.77But yet this is true, that looke what pro∣portion each of these seuerall sums doth beare to 21, the same proportion doth the iust char∣ges of euerie man (beside his ouerplus) beare to the totall of the charges, the ouerplus be∣ing deducted: wherfore this maye you note, that before you do apply the total of the char∣ges to the Golden rule: you must deducte the ouerplus whiche is 6, 12, and 20, that is in the whole 38 but then 8 must be restored for the abatement of the thirde man, and then remaineth to be deducted 30, Take 30 ther∣fore out of 3000, and there will reste 2970, whiche I must set in the Golden rule for the seconde summe: and for the third summe I must put ech of the smal numbers before mē∣tioned, whiche although they be not the seue∣rall charges, yet they represent them in pro∣portion. And so making for euerie mannes

charge a seuerall question, the figures wil be 4, which I marke with foure letters, a. b. c. d. thus.

〈 math 〉〈 math 〉

Where I haue set for briefnesse the summe of euerie mans charge in the fourthe place, presupposing that you can tel how to trie out that fourth sum by so many examples as yée haue had.

Sch.

As I truste that I vnderstand thys fourme, so I desire muche to knowe what maye be saide for thē that mistooke this que∣stion.

Maister.

You séeme so desirous to know this errour, that you haue forgotten to examine whether this worke bee wythoute

Scholer.

Mée séemeth this worke to bée well done, bycause the Addition of the 4 se∣uerall numbers doth make the totall summe

of 2970, whiche was to be diuided into suche foure parts.

Maister.

But then haue you forgotten that the firste man must pay 6 crownes more be∣sides this share, and the seconde man twelue crownes more: the third mā 8 crownes lesse: and the fourthe man 20 crownes more, for without these, your first total of 3000 crow∣nes wil not be made.

Scho.

Then must I adde to the first mans summe 6 more, and it wil be 854 4/7: and to the second summe I must adde 12, and it wil be 577 5/7: from the thirde summe I muste a∣bate 8, and then wil the sum be 1123 3/7: then adding vnto the 4 summe 20, it will be 444 2/7: and these 4 sums will make 3000, whiche is the whole charge, as in this example it maye appeare, where firste I 〈 math 〉〈 math 〉 gather the 14/7. that maketh ••, and so procéede I in the Ad∣dition to the end.

Ma

Nowe haue you wel done, and this worke in the same sums is brought of other learned men for the true solution of the question as it was first proponed, which as (I saide) was impos∣sible: and nowe examine it by these seuerall

summes, and sée whether it do agrée with the summes in the question proponed.

The first man must pay 1/•• and 6 ouer of the total sum: how thinke you, is 845 4/7 the halfe and 6 more of ••000:

Sc.

No that it is not, for it would be 1506: and for the second man 101••: and for y^e thirde man 1992: and for the fourth mā 770, wher∣of not one summe agréeth to this worke. But I maruaile that so wise mē could be so much ouerséene.

Ma.

It is commonlie séene, that when men wil receiue things from elder writers, and will not examine the thing they séeme ra∣ther willing to erre with their auncients for companie, than to be bolde to examine theyr workes or writings, which scrupulosity hath ingendered infinite errours in all kindes of knowledge, and in all ciuill administration, and in euerie kinde of arte: but these learned men did not meane anye other thing by this question, than to finde suche numbers as shoulde beare the same proportion togither, as those numbers in the question proponed did beare one to an other: whiche thing you shal perceiue more plainelie by an other que∣stion of theirs, that is this.

* 1.78A man lying vpon his death bed, bequea∣theth his goods (whiche were worthe 3000 Crownes) in this sorte. Bycause his wife was great with childe, and he yet vncertain whether the childe were a male or female, he made his bequest conditionally, that if his wife bare a daughter, then should the wife haue halfe his goods, and the daughter ⅓, but if she were deliuered of a sonne, then that son should haue ½ of the goods, and his wife but ⅓. Nowe it chaunced hir to bring forth both a son and a daughter, the question is: How shal they part the goods agréeable to the testatour his wil.

Scholer.

If some cunning Lawyers hadde this matter in scanning, they woulde deter∣mine this Testamente to be quite voide, and so the man to die intestate, because the testa∣ment was made insufficient, sith this condi∣tion was not expressed in it, and also it might haue chaunced that she shoulde haue brought forth neither sonne nor daughter, as often hath bene seene: so is the will vnsufficiente in that point also.

Maister.

Suche scanners shoulde séeme to cunning, and yet not so cunning as cruel: for the minde of the Testatour is to be taken fa∣uorablie,

for the aid of y^e legatories whē there riseth such doubts. But let vs trie this work, not by force of lawe, but by proportion Geo∣metricall, séeing the testatour did minde to prouide for each sort of them.

Sc.

If the sonne shal haue ½ by force of the Testament, so must the mother haue ⅓. Again because she hath a daughter also, therefore ought she to haue ½ and the daughter 1/••: that is both wayes ½ ⅓, and ½ ⅓, whiche commeth to the whole goods, and ⅔ more. Wherefore it séemeth also impossible.

Maister.

In this matter the minde of the Testatour is so to be vnderstande, that suche proportion shoulde be betwéene the portion of the wife and the sonne, as is betwéene ⅓ and ½ that is, the sonne muste haue ⅙ for 2/6 to his mother, so shall he haue •• to 2, that is as much as his mother, and half as much more: and the mother muste haue the like rate in comparison to hir daughter. Then muste I finde out 3 numbers in suche proportion, that the firste may be as muche as the second, and halfe as muche more (that is) in proportion sesquialtera, and the seconde to the thirde in the same proportion, suche numbers be 6, 9, 4.

Sc.

I pray you sir, how shal I find out those numbers?

M.

That wil I gladly tel you.

* 1.79Whatsoeuer the proportion be of any thrée numbers, multiplie the Termes of that pro∣portion togither and the number that amoū∣teth, shall be the middle number of the 3: thē multiple that middle number by the lesser tearme, and diuide that totall by the greater, and the least number of the 3 will amounte. So if you multiplie that middle number by the greater extreame, and diuide that total by the lesser extreme, then wil the greatest num∣ber of that progression amount.

Scholer

* 1.80Then in this example, to finde the proportion of ½ to ⅓, I muste diuide (as you taught me in Diuision) ½ by ⅓, & the quotient will be ½, that is 1 ½, whereby I perceiue that the proportion in this question is, as 3 to 2. Therefore (as you taughte me euen nowe) I multiplie 3 by 2, and the summe is 6, which must be the middle number: then I multiply the middle number 6 by 2, which is the least terme, and the summe is 12, that doe I diuide by 3, being the greater Terme, and the quotient is 4, so is 4 the leaste num∣ber of the 3. Then I multiplie 6 by 3, wher∣of

commeth 18, and that I diuide by 2 and so haue I 9, which is the greatest number of the 3.

Ma.

An other way yet maye you finde the third number in any progression, if you haue 2 of them: for if the middle number be one of them which you haue, then multiplie it by it self (as in this example 6 by 6 maketh 36) and that totall diuide by the other number which you haue, and the third number will be the quotient.

Sc.

Then if I diuide 36 (which commeth of 6 multiplied by it selfe) by 4, the quotient will be 9, & if I diuide 36 by 9, the quotient will be 4. But what if I knowe the first nū∣ber and the thirde, and would haue the mid∣dle number?

May.

Multiplie the two numbers togi∣ther, and in their totall you must séeke the roote of that number, and it shall be the mid∣dle number: but because as yet you haue not learned how to extract rootes, therfore vse the first forme which I haue taught you, till I teache you to extract rootes. And now go for∣warde with the aunswere to the same que∣stion.

Scholer.

I perceiue then that the son muste not haue ½ of the goods, neither the mother ⅓, nor yet the daughter ⅓, but yet muste the goods be diuided in suche proportion, that the sonne shall haue 9 crownes for 6 to hys mo∣ther: and the mother shal haue 6 crownes for euerie 4 to the daughter. Then I applie it to the Golden rule in thrée 〈 math 〉〈 math 〉 examples thus: where the firste number is the Addi∣tiō of those thrée numbers 9, 6, 4 and the thirde is one of them seuerallie: the seconde is the totall of the goods in the testament: & then by the worke of the Golden rule I finde out y^e fourth number in euerie work that is for the sonne 1705 1/11: for the mother 〈 math 〉〈 math 〉 1136 16/15: and for the daughter 757, 17/15, which thrée summes added togither do make the sū of the whole goods, as may be séene by this example.

And this (me thinketh) I doe perceiue, that bicause in this case ther is a necessary remedy diuised againste an vrgente inconuenience,

therfore those lerned mē thoght they might vse the like liberty in that other question.

Maister

Your gesse is good, but they had so good reason for them in the one, as they haue in the other: as in another example of theirs, it may better appeare, that is this:

A man left vnto his iij.* 1.81 sons 7851 crow∣nes, to be parted in this sorte, that the firste sonne should haue ½, the second sonne ⅓ and the third sonne ¼, which is not possible, for ½ ⅓ ¼ doth make 2/2 0/4: or 1/1 ••/2;, that is 1 ••/12, so is it more than the whole: but reduce these fra∣ctions into one denominatiō, the least that they wil come to, and they wil be 6/12, 4/12, 3/12, and so may you part the goods in such pro∣portion as these 3 Numeratours beare to∣gither: that is, the firste to haue 6 for euery 4 to the seconde: and the seconde to haue 4 as oftē as the third hath 3: and so their por∣tions wil be for the firste, 2623 7/13: for the second 2415 0/13: and for the thirde 1811 1••/13, and those 3 shares added togither, 〈 math 〉〈 math 〉 will make the totall summe of the whole goodes, as you may easily sée in this example Another question is there pro∣poned thus:

There is 450 Crownes to be diuided be∣wéen 3 men, so that the first man must haue ½ ⅓, the second man ⅓ ¼, the thirde man shall haue ¼ ⅕.

Scholer.

I maruaile that anye manne should be so ouerseene to propound that que∣stion as a thing possible, sith ½ ½▪ 1/•• ¼▪ ¼ ⅕, doe make 1 ⅓ ⅕, that is almost double the whole summe.

But I perceiue it might be thus proponed, that as often as the first man did receyue 50 Crownes, so often the second man should re∣ceyue 35, and the third man 27, for ½ ⅓, is equall to 50/60, and so is ⅓ ¼ equall to 1/6 5/••, and ¼ ⅕ is 2/6 7/••, and so working the question, the 3 figures will appeare in this forme: where∣by the firste mannes 〈 math 〉〈 math 〉 portion is founde to bée 200 50/56: the se∣conde mannes parte is 140 15/56: the third mans share is 108 24/76: which in y^e whole doth make 450 crou∣nes y^t was the whole summe to be diuided betwéene them.

Mayster.

And thus you are (I thinke) sufficiently instructed in the rule of Felow∣ship.

The Rule of Alligation.

NOw will I go in hād with the rule of Alli∣gatiō,* 1.82 which hath his name, for that by it there are diuers par∣cels of sundrie prices, and sūdry quantities alligate boūd or mix∣ed togither, whereby also it mighte be well called the rule of mixture, and it hath great vse in composition of medicines, and also in mixtures of metals, and some vse it hath in medicines of wines, but I wish it were lesse vsed therein than it is now a dayes. The or∣der of the rule is this:* 1.83 When any summes are proponed to be mixed, set them in order one ouer an other, and the common number wherevnto you will reduce them, set on the left hand, then marke what summes be lesser than that common number, and whiche bée

greater and with a draught of your perme, e∣uermore linke 2 numbers togither, so that one bee lesser than the common number, and the other greater than hee, for two greater or two smaller can not well be linked togither, and the reason is this, that one greater & one smaller may bee so mixed, that they wil make the meane or common number very wel, but 2 lesser can neuer make so many as the com∣mon number, being taken orderly: no more can two summes greater than the meane, ne∣uer make the meane in due order as it shal a∣peare better to you hereafter. And as it is of necessity to lynke euery smaller (once at the leaste) with one greater, and euerye greater with one smaller: so it is at libertie to linke them oftner than once, and so may there bée to one question many solutions. When you haue so linked them, then marke how muche eche of the lesser numbers is smaller than the meane or commō number, and that differēce set against the greter numbers which be lin∣ked with those smaller eche with his match still on the right hande, & likewise the excesse of the greater numbers aboue the meane, you shall set before the lesser numbers which bée combined with them.* 1.84 Then shall you by Ad∣dition

bring all these differences into one summe, which shal bée the first number in the Golden rule: and the seconde number shal be the whole masse that you wil haue of al those particulers: the thirde summe shal be ech dif∣ference by it selfe and then by them shall bée founde the fourth number, declaring the iust portion of euery particular in that mixture. As nowe by these examples I will make it playne.

There is foure sorts of wine of seueral pri∣ces,* 1.85 one of 6 d a gallon, an other of 8 d the thirde of 11 d, and the fourth if 15 pens the galon, of al these wines would I haue a mix∣ture made to the summe of fiftie galons, and so that the price of eche galon may be 9 pens. Now demande I howe much must bée taken of euery sort of wine?

Scholer.

If it shall please you to worke the first example, that I may marke the ap∣plying of it to the rule, then I trust I shall be able not only to doe the like, but also to sée reason in the order of the worke.

Maister.

Marke then this forme and the placing of euery kinde of number in it.

* 1.86〈 math 〉〈 math 〉

Here (you sée) I haue sette downe the seue∣rall prices which be 6, 8, 11, 15, and haue linked together 6 with 15, and 8 with 11. The common price 9, I haue set on the lefte side: And the difference betwéene it, and e∣uery particular price, I haue set on the right hand not against the summe, whose differēce it is, but against the summe that it is linked withall: so the difference of 15 aboue 9, is 6, which I haue set not against 15, but against 6, that is linked with 15, and the difference betweene 6 and 9 (that is 3) I haue set a∣gainst 15. So likewise y^e difference betwene 8 and 9 is but 1, that haue I set against 11, and y^e difference of 11 aboue 9 (which is 2) I haue set against 8. Then adde I all those 4 difference, and they make 12, which I sette for the first number in the Golden rule: the seconde number I make 50, which is the

summe of gallons that I woulde haue, and the thirde summe is euerie particular diffe∣rence. Nowe if you worke by the Golden rule, you shall finde the number of Gal∣lons that shall be taken of eache sorte of wine: For the better distinction whereof, I haue sette these letters abcd both agaynste the numbers for which the workes do serue, and ouer the workes also, whiche seuerallye serue for eache of them. And nowe if you liste to examine the truthe of these workes, adde those foure summes togither,* 1.87 and they wyl make fiftie, that is the 〈 math 〉〈 math 〉 totall whiche I would haue, as by this exam∣ple you may easily per∣ceiue. And for to proue how y^e prices do agrée, doe this. Multiplie this totall summe 50, by the common price 9, and it will make 450: then kéepe that summe by it selfe, and after∣ward Multiplie euerie seuerall summe of Gallons, by the price belonging to the same Gallons, and if that summe doe a∣grée with this, whiche you haue kepte firste,

then is your worke well done. As here, 25 is the number of gallons of 6 d price, multi∣plie then 25 by 6, and it maketh 150, whiche you shall set downe: 〈 math 〉〈 math 〉 then multiplie 8 2/6 by 88 which is the price for y^e number of Gal∣lons, and it will make 66 4/6: so again 4 ¼ multiplied by 11 doth make 45 5/6. And last of al 12 3/6 multipli∣ed by 15, maketh 187 ⅙. And these added to∣gither doe make 450, as in the example an∣nexed you may sée: wherefore séeing it doth a∣grée with the former sum of 50, multiplyed by 9, I may iustlie affirme this worke to bée good and wel done.

* 1.88And now to proue how you can do the like, I propounde the same question, only willing you to vse some other forme of combining or linking the summes.

Scho.

That shall I proue with your fa∣uoure, and therefore I combine 8 with 15, and 6 with 11, and then the forme wyll be as foloweth:

〈 math 〉〈 math 〉 whereby amounteth the same summe in to∣tall of the differences, as did before: and yet nowe the differences be altered, as the combi∣nation is chaunged, whereof I vnderstande the reason by your former worke. And ther∣fore here appeareth no straunge thing, but that nowe I muste haue 8 2/6 gallons, of sixe pence and 25 gallons of 8 d and 12 gallons and ⅙ of 11 d, and so conse∣quently 〈 math 〉〈 math 〉 4 gallons and ⅙ of 15 d, so y^e multiplying 8 2/6 by 6, it maketh 50, and thē 25 multiplied by 8, maketh 100: likewise 12 3/6 multi∣plyed by 11, yéelded 137 3/6, and 4 ⅙ multiplied by 15, maketh 62 3/6, whi∣che foure summes added in one, will yéelde in the totall 450, whiche agréeth wyth the Multiplication of 50 (being the total summe

of gallons) by 9 the common or meane price.

Mai.

Séeing you conceiue this worke so wel, I wil propound an other example vnto you of more varietie in the Alligations or cō∣binings: as thus.

* 1.89A Merchaunt being minded to make a bar∣gaine for spices in a mixte masse, that is to say, of Cloues, Nutmegges, Saffron, Pep∣per, Ginger, and Almonds, the Cloues being at 6 s apounde, the Nutmegges at 8 s. Saf∣fron at 10 s. Pepper at 3 s. Ginger at 2 s. and Almonds at 1 s.

Now woulde he haue of eche sorte some, to the value of 300 lb, in the whole, and ech poūd one with an other to beare in price 5 s. howe much shal he haue of ech sort?

Sch.

That wil I trie thus.

First I set down those sixe seuerall prices, and at the left hand I set the common price 5 s. Then I linke thē thus, 1. with 10, ••, wyth 6, and 3 with 8. As in the example follow∣ing.

〈 math 〉〈 math 〉

Maister.

I hadde minded to haue combi∣ned them in more varietie, but I am content to sée your owne worke firste, and then more varieties in combination maye followe a∣none.

Scholer.

Then to continue as I beganne, I séeke the difference betwéene 1 and 5 (which is 4) and that I set against 10: then againste 1 I set 5, whiche is the excesse of 10 aboue 5: so I gather the difference betwéene 2 and 5, whiche is 3, and that I set against 6, bicause it is combined with 2: and likewise the dif∣ference of 6 aboue 5 (which is ••) I set against 2. Then take I the difference of •• from 5, whiche is 2, and that I set against 8, and be∣fore that 3 I set the difference of 8 aboue 5, whiche is 3. Then gather I al these differen∣ces

by Addition, and they make 18, which I set for my first number in the Golden rule, & so appeareth by those workes, y^t of Almonds I must take 83 lb ⅓, of Ginger 16 lb ⅔, of pep∣per 50 lb, of Cloues 50 pound, of Nutmegges 33 pound ⅓ and of Saffron 〈 math 〉〈 math 〉 66 pounde ⅔. Then for try∣all hereof, I multiplie eue∣rie parcell by his seuerall price, as 83 1/•• whiche is the summe of Almondes, I multiplie by 1, whiche is their price.

Also 16 ⅔ the summe of Ginger I multi∣plie by 2, which is the price of it. And so each other in his kind, as this table annexed doeth represent: and then adding them altogither, I finde the total to be 1500, which also will a∣mounte by the multiplication of the grosse masse of 300 by the common price 5, where∣fore it appeareth wel wrought.

Ma.

Nowe will I make the Alligation to proue your cunning somewhat better: but bycause you shall not thinke your selfe pres∣sed so much, I will also note the differences, as in this example you may sée.

〈 math 〉〈 math 〉 where I haue alligate 1 with 6 and 8: and therefore haue I set against 1, both their dif∣ferences: that is 1 and 3. Likewise bycause 2 is combined with 8 and 10, I set before hym their differences, 3 and 5. Againste 3 I haue sette onlie 5, whiche is the difference of 10, with whome 3 is combined onelye: like∣wise 6 is onelie Alligate to 1, and therfore is the difference of 1 onlie set againste it: 8 is lincked with 1 and 2 and therefore hathe hée against him both their differences, 4 and 3: and 10 is ioyned with 2 and 3, therefore hath be their differences 3 and 2. And because of ease for you, in an other columne I haue set the differences reduced into one number, for

euerie seuerall sort, and haue also added thē togither, wherby appeareth that they make 33, and so consequently you sée the workes of the golden rule set forth for y^e sixe seueral drugges: I haue added letters a, b, c, &c as before. But I would not wish you to cleane stil to these elemētary aides, but accusiome memorie to trust hir self, so shal occasion of negligence be best auoyded. And as for the proofe, trie it at more leysure, because y^t time now is short, and you sufficientlie instruc∣ted in that proofe. And there resteth diuerse things behinde yet, of whiche I woulde gladlie giue you some taste before our de∣parture.

Scholer.

But if it may please you to lette me sée al the variations of this question, be∣fore you go from it, for me thinketh I could varie it two or three wayes more yet.

Ma.

I am content to sée you make two or thrée variations, but I would be loth to stay to sée al the variations, for it may be varied aboue ••00 wayes although manye of them would not wel serue to this purpose.

Sc.

I thought it impossible to make so ma∣ny variations.

Ma.

Meruaile not therat, for some questi∣ons

of this rule may be varied aboue 1000 wayes, but I woulde haue you forget suche fantasies, till a time of more leysure. And now go forward with some variation of this question.

Scholer.

For the first variation, I linke the first number 1 with 8 and 10, and 2 I combine with 6 and 10, then ioine I 3 wyth 6, 8, and 10, as in this forme.

〈 math 〉〈 math 〉

And so doeth there appeare the portion of weight for euery kind of drugge in this mix∣ture. Now for the triall.

Maister.

Nay stay there, you shal not néed to make triall in one example so often, or if you liste to do it by your selfe, I am content. But nowe sette foorth (for declaration that you conceiue the rule) two or thrée examples of seueral combinations, and then will wée

and then will we passe to some other exam∣ple and so ende this rule.

Scholer.

As it pleaseth you so will I doe. And these be the varieties in whiche as the 〈 math 〉〈 math 〉 combinations are seuerall, so doth it plainely appeare, that the differences by whyche the proportion of eache seuerall kinde is taken, are also seuerall. And yet I sée in the thrée first of these fiue varieties, and in one other, before, the totall summe of the differences to

be one, that is to say 18, whereby I perceiue that the varietie of their mixture doth depend of the varietie of their differences seuerall, and not of the varietie of their totall summe.

Maister.

So is it. And séeing you con∣ceiue it so well, I will make an ende of thys rule, onely exhibiting to you one question or two of the mixture of metalles, that by it you may deuise other like, and exercise your self therein also, bicause the vse of it serueth often in businesse of charge, not so much for Gold∣smithes, as for coynage in mintes. Firste I demaunde of you this question. If a Minte∣maister haue golde of 22 karectes, and some of 23 karectes, some of 24: Againe, some of 15, some of 16, and some of 18 karectes, and would mixe them so, that he might haue 100 ounces of 20 karects, howe much shal he take of euery sorte?

Scholer.

To knowe that, I aunsweare in order thus.

〈 math 〉〈 math 〉

Maister.

You haue wrought the question well, but howe chaunced you made no doubt of that newe name, Karecte?

Scholer.

Because I thoughte it out of time to demaund suche questions now, séeing you make so much haste to ende: and againe in this case the proportion of the numbers is sufficient for my purpose in this worke, tru∣sting, that an other time you will instructe me as wel of this, as of sundry other things, which I haue heard you talkē of, so I haue a great desire to knowe them.

Maister.

Your aunsweare is reasonable: and your request and trust with Gods helpe I intende to satisfie. And to goe forwarde with this matter, let me sée your examinati∣on of this last worke.

Scholer.

First for the one part I adde to∣gither

all the particular summes as 〈 math 〉〈 math 〉 they appeare in the worke, and they make 100, as here by their Additi∣on it doth appeare.

And so it seemeth, that the sum∣mes are well gathered, but for the farther triall of them, I multiplye first 20. which is the common or meane sum 〈 math 〉〈 math 〉 of the karectes by 100, whiche is the summe of the whole masse which I woulde haue and it ma∣keth 2000. Then I multiply e∣uerye particular summe by the karects that it doeth containe, as 10 by 15, and that maketh 150.

Likewise I multiplye 15 by 16, and it yéeldeth 240: so 20 by 18 maketh 360. And 25 by 22 yéeldeth 550: likewise 20 by 23 bringeth forth 460: and last of al 10 multiplyed by 24, yéeldeth 240: whiche summes all ioined togither make 2000, that doeth agree with the like summe before: wherefore I may well saye, that the worke is good. And nowe if it please you I would sette forth some varieties of this que∣stion, to proue my witte.

Maister.

Goe to, let me sée.

Scholer.

Here be foure varieties.

〈 math 〉〈 math 〉

And more yet I coulde make, but not like is the number that you spake of in the varia∣tion of the other question.

Maister.

That will I teache you at more leisure, séeing it is a thing rather of pleasure, than of any necessitie.

But nowe for youre exercise in this rule, one other question I will propose. A minte maister hathe 6 ingottes of siluer of sundrie finenesse, some of 4 ounces fine, and some of

5 ounces, some of 6, and other of 8, some of 11, and other of 12: and his desire is to mixe 500 pound weight, so that in the whole masse euery pound weight shoulde beare 9 ounces of fine siluer, howe muche shall he take (saye you) of euery sort of siluer?

Scholer.

To finde 〈 math 〉〈 math 〉 out that I sette the numbers thus in or∣der.

And gathering the differēces, it will appeare, that of the firste sort there must be 43 ½ ½ of the secōde like muche: of the third sorte, 65 5/23: and of the fourth sorte as muche: of the fifth sorte 195 ½ 5/••: and of the sixth sorte 86 2/2 ⅔, whiche in the whole wyll make 300 lb waight: and in ounces after 9 ounces fine 4500, that is of the firste sorte 173 2/2 ½: and of the second sorte 217 9/23: of the third sorte 391 7/23: of the fourth sorte 521 ½ 7/3: of the fifth sorte 2152 4/2 ••/3, and of the sixth sorte 1043 ½ ⅓, whiche al togither doe make 4500 ounces, agréeable to the multiplication of 9 by 500.

Maister.

This is well done of you, there∣fore nowe make thrée or foure variaties, and so an ende of this rule.

Sc.

These 4 varieties I set for example.

〈 math 〉〈 math 〉

Maister.

And by these it appeareth, that you can find out more, with which I wil not nowe meddle, saue onely for to shewe you an easie help in drawing the lines of Combina∣tion, I wil set forth two varieties here.

〈 math 〉〈 math 〉

And this shall suffise now for the rule of Alligation or mixture, for by these examples may you easily coniecture suche other as doe appertaine to it, as wel for the due working, as for variety of drawing the lines of combi∣nation.

Scholer.

Sir, albeit it pleased you ere while, to put me from my musing at the ma∣nifolde varieties, that may fall in these com∣binations, and termed them fātasies, yet my fantasie giueth me, that the consideration of this shoulde in many other examples and ca∣ses of importance be very néedefull, and the knowledge of it most profitable. Therefore ye may wel thinke, that at another time con∣uenient I will request you to aide me héere∣in.

Maister.

Truth it is, that this considera∣tion may fall in practise as wel Politicke, as Philosophicall, and sundrie wayes in them

be applyed, therefore when time shall fall •• for the discussing of this consideration, you shal not want my helping hande.

The rule of Falsehode.

NOwe wil I briefly al∣so teach you somwhat of the rule of Falshod, whiche beareth hys name,* 1.90 not for that it teacheth any fraude or Falshoode, but for that by false numbers ta∣ken at all aduentures, it teacheth howe to finde those true numbers that you séeke for.

Scholer.

So might any other rule be cal∣led the rule of Falshoode, for they worke by wrong numbers, and by them finde out the right numbers, so doth the rule of Alligation, the rule of Felowship, and the Golden rule partely.

Maister.

In the Golden rule, the rule of

Felowship, & the rule of Alligation, although the numbers that you worke by, be not the true numbers that you séeke for, yet are they numbers in iuste proportion, and are founde by orderly worke: whereas in this rule, the numbers are not takē in any proportion, nor founde by orderly worke, but taken at all aduentures.

And therefore I sometimes being merry•• with my friends, and talking of suche questi∣ons, haue caused them that proponed suche questions to call vnto them suche children or ideots, as happened to be in the place, and to take their answeare, declaring, that I would make them solue those questions, that semed so doubtfull.

And indéede I did answeare to the questi∣on and worke the triall thereof also, by those answeares which they happened at al aduen∣tures to make: which numbers séeing they be taken as manifest false, therefore is this rule called the rule of false positions, and for briefnesse, the rule of Falshode, which rule for readinesse of remembrance, I haue com∣prised in these few Verses following, in form of an obscure Riddle.

Gesse at this worke as happe doth leade, By chaunce to truth you may proceede. And firste worke by the question, Although no truth therein be done. Such falsehoode is so good a grounde, That truth by it wil soone bee founde. From many bate to many mo, From to fewe take to fewe also. With too much ioine to fewe againe. To too fewe adde to many plaine. In crosse waies multiply contrarie kinde, All truth by falsehoode for to finde.

The sense of these Verses, and the summe of this rule, is this:

When any question is proponed apper∣taining to this rule, firste imagine any num∣ber that you liste, whyche you shall name the firste position, and put it in stéede of the true number, and then worke with it as the que∣stion importeth: and if you haue missed, then is the laste number of that worke, eyther too greate or too little: that shal you note as here∣after shall be taught you, and you shall call it the firste errour.

Then beginne againe, and take an other number whiche shall be called the Seconde

position, and worke by the question: if you haue missed again, note the excesse or default as it is, and call that the second errour. Then multiply crosse waies the first position by the second errour, and againe the seconde positi∣on by the firste errour, and note their totalls seuerally by the names of Totalles. Then marke whether the two errours were bothe like, that is to say, both too much, or bothe too little: or whether they be vnlike, that is, the one too much, and the other too little, for if they be like, then shall you subtract the one totall from the other (I meane the lesser from the greater) and the Remainer shall be your diuidend, so muste you abate the lesser error out of the greater, and the residue shal be the diuisour. Now diuide the diuidend by that diuisour, and the quotient will shew you the true number that you séeke for: But and if the errours be vnlike then must you adde both those totalles (whych you noted) togy∣ther and take that whole number for the di∣uidend, so shall you adde both the errours to∣gither, and that whole number shall be the diuisour, and the quotient of that Diuision shall giue you the true number that the que∣stion séeketh for: and thys is the whole rule.

Scholer.

This rule seemeth so vnlike any other, that without some example I shal not easily vnderstande it

Maister.

Therefore take this example: A Mason was boūd to build a wall in 40 days, and it was couenaunted so with him, that euery daye that he wrought, he shoulde haue for his wages 2 s, 1 penny, & euery daye that he wrought not, he shoulde be amerced 2 s, 6 d, so that when the wall was made, and the reckning taken of the dayes that he wrought, and of the other that he wrought not, the Ma∣son had clearely but 5 s, 5 d, for his worke. Nowe doe I demaund, how many dayes did he worke of those 40, and how many did hée not worke?

Scholer.

I praye you expresse the order of the worke, that I may partely by imitation, and partly by comparing it with the rule, bée able againe to do the like.

Maister.

This order shall you kéepe in the worke of this rule: firste take some number (as you list) at aduenture, as for example: I saye he played 12 dayes, and wrought 28 dayes. Nowe caste you the wages of euerye daye, and sée whether it will agrée with the summe of 5 s, 5 d.

Scholer.

The 28 dayes that he wrought after 25 pence the daye, yéeldeth 700 d. Then the 12 dayes that hée wrought not, at 30 pence eche daye, doth amount to 360 pence, whiche if I abate out of 700 pence, there resteth 340: but you say he had not so much.

Maister.

He hadde but 65 pence, and by this supposition he shoulde haue hadde 340: therfore is this summe too much by 275 whi∣che summe I muste set downe after this sort, as you sée here, where firste I 〈 math 〉〈 math 〉 haue made a crosse (common∣lye called Sainct Andrewes crosse) and at the ouer corner on the left hand I haue sette the firste position 12, and at the other corner vnder it, I haue set 275 which is the first er∣roure, with this figure 4, whyche betoke∣neth too muche, as this line, — plaine without a crosse line, betokeneth too little. On the right hande of the crosse I haue lefte two like rowmes for the second position and his errour. Therfore to prosecute the worke, I suppose he played 16 dayes, and wrought 24.

Scholer.

I was a while in doubte why

you named the dayes of hys working, séeyng they be not set in the figure: and I doubted howe you knewe them, or else whether that you did suppose them at all aduentures, as you didde the dayes that hee played: but now I gather, that seeing 40 dayes is the whole time limited, then the dayes that hée played being supposed the rest of 40 muste néedes be the dayes that he wrought, and therefore 28 followed 12 of necessitie and 24 foloweth 16 also of necessitie: but yet I scarce perceyue why you set not in the figure as well 28 as 12.

Maister.

It forceth not whiche of them I take,* 1.91 so that in the seconde position I take the numbers of the same nature that is here bothe of working dayes, or bothe of ydle: but nowe examine you this seconde positi∣on.

Scholer.

If he played 16 dayes, then aba∣ting 16 times 30 d the summe will be 480 d. And for 24 dayes that he wrought, euery day yéelding 25 d, the totall is 600 d: so that abating 480 out of 600, there resteth 120: and as you say it shoulde be but 65, therefore it is too much by 55, that must be sette on the right hand of the figure at the neather parte,

and ouer it on the same 〈 math 〉〈 math 〉 side 16 which is the se∣conde position, thus. And as I gather by your words, it were al one if I did set 28 in steade of 12, and 24 in steade of 16.

M.

So were it. But this shall you marke,* 1.92 * 1.93 that of what nature so euer the two positions be, of the same nature is the quotient. There∣fore when the positions in this question are 12 and 16,* 1.94 which both being numbers of the playing dayes, the quotiente shall declare the true numbers of playing dayes, where as if the positions had bene 28 and 24, whyche are supposed to be the working dayes, then woulde the quotient declare the true number of the working dayes, & not of playing dayes as it will doe nowe. And therefore to con∣tinue the worke of this question, and to find the true number of playing dayes, I muste multiplie crosse wayes the firste position 12 by 55, that is the seconde erroure, and the totall will be 660, then I multiplie 275 by 16, and it yéeldeth 4400. Now bicause the er∣rours are like, that is to saye, both too much, I muste subtracte 660, out of 4400, and so

remaineth 3740, whiche is the diuidend. A∣gaine I must subtract the lesser errour 55 out of 275 that is the greater errour, and there wil remaine 220, which shall be the diuisor, then diuiding 3740 by 220, the quotient wil be 17. Wherefore I saye nowe constantlie, that 17 is the true number of dayes that the Mason played: and then it followeth, that he wrought 23 dayes, and so is the question an∣swered.

Nowe for the order of triall of this worke there néedeth none other proofe but only thys, to worke with this number according to the question, and if it agrée, then appeareth the number to be it that you would haue. As here nowe séeing he wrought 23 daies, and muste haue for euerie day 25 pence, the whole sum commeth to 575. Then again séeing he play∣ed 17 dayes, and muste abate 30 pence for e∣ueriedaye, the whole summe of the abate∣ment will be 510: therefore I subtracte 510 out of 575, and there wil remaine 65, which maketh 5 s, 5 d. the cleare wages of the Mason for his worke, according to the que∣stion.

Scholer.

Nowe I trust I vnderstand the worke and the rule so well (and the better by

thys proofe) that I can be able to do the like. And for a proofe I take the same question all saue the laste number, where I will suppose that he hadde 10 s, for his wages cléere. And now to gesse at the number of y^e days that hée wrought, I suppose first that he wrought: 20 dayes, then say I, if he wrought 20 dayes his wages must be 500 d, then did he play other 20 dayes, for whych must be abated 600 d, and then he leeseth 100 d. And so am I at a staye, for it is not like vnto your former worke.

Ma

You shoulde haue required of me some question, and not haue taken a question of your owne fantasying, vntill you were more expert in this arte:* 1.95 for so might you as well happen on an impossible question as on a pos∣sible: but now to go forwarde, consider that this number is too little by 220, séeing he should gaine by your supposition 120 d, and in this positiō he léeseth 100, those both make 220, whiche you shall set downe for the first error with this signe—, betokening to little, as here in this forme following doeth ap∣peare.

And nowe for the rest go for∣warde 〈 math 〉〈 math 〉 youre selfe once a∣gaine,

Scholer.

As my errour hath vttered my follie, so it hath procured me better vnderstanding. Nowe therefore consi∣dering this position not to solue the question, I take an other, supposing that he wroughte 30 dayes, then for his wages he must be allo∣wed 750 d, and for the 10 dayes whyche hée wrought not, he muste abate ••0•• d, and so remaineth cleare 450 d: but it should be one∣ly 120 d, therfore is it to much by 330, whi∣che I sette downe in the figure with the for∣mer position, and his erroure and the figure appeareth thus.

Nowe must I mul∣tiplie 〈 math 〉〈 math 〉 in crosse wayes 220, by 30, and it will be 66000. Then againe I multiplie 330 by 20, and it will be also 6600. Where∣fore if I shall subtract the one out of y^e other, there will remaine nothing to be the Diui∣dend.

Ma.

In this you forget your selfe again: for in as much as the signes in the errors be vn∣like,

therefore must you worke by Addition, adding togither those two totals to make the diuidend, and also adding the two errours to make the diuisor. And bycause you shall no more forgette this part of the rule, take this briefe remembrance:

Vnlike require Addition, And like desire Subtraction.

Sch.

You meane, that if the errours haue like signes, then muste the diuidend and the diuisor be made by Subtraction, as is taught before: And if those signes be vnlike (as in this laste example they be) then muste I by Addition gather the Diuidend and the diui∣sor. Therefore must I adde 6600 to 6600, and it will be 13200, whiche shall be the di∣uidend. Then againe I adde 220 to 330, and it will be 5••0, whiche muste be the diuisor: wherefore diuiding 13200 by 550, the quo∣tient will be 24, whereby I knowe that the Mason wrought 24 dayes: and then it follo∣weth that he played 16 dayes.

M.

Examine your worke whether it bée a∣gréeable to the question or no.

Sc▪

For 24 dayes worke, the wages muste

be 600 d. and for 16 dayes whiche the Ma∣son wrought not, there muste be abated 480, and then remaineth cleare to the Mason 220 pence, as the question importeth, wherefore it is euident, that 24 is the true number of the dayes that he wrought.

Maister.

Althoughe you séeme nowe to vn∣derstande this worke, yet to acquainte youre minde the better with the newe trade of this rule, I thinke it good to propone to you fiue or sixe examples more, before I make an end of it.

Sch.

Sir I thanke you, that you do so consi∣der my commoditie and profit in knowlege, for vndoubtedlie it is practise & exercise that maketh men prompt & expert in euerie kinde of knowledge.

Maister.

You say well so that they follow some certaine preceptes to gouerne and rule their practise by, else maye practise procure custome of error, and a repugnance to exact∣nesse of knowledge, namelie as long as the error is not plainelie knowen to the vulgare sorte. But to returne to our worke. There is a seruant that hath boughte of the veluet and da∣maske for his maister 40 yeardes, the veluet at 20 s, a yeard, and the Damaske at 12 s, &

when he commeth home,* 1.96 his maister deman∣deth of him howe muche he hath boughte of ech sort: I cā not tel (saith he) exactlie, but this I knowe, that I paide for damask 48 s. more than I paide for veluet, nowe must you gesse how manie yeardes there is of each sort.

Scho

Although the gesse séemeth difficulte, yet I wil proue what I can do: for I remem∣ber your saying, that it forceth not how fonde or false the gesse be, so it be somewhat to the question, and not an answere of a contrarie matter.

Therefore first I imagine that he boughte 20 yeardes of Damaske, for which he should paye after the former price 240 shillings: then muste he néedes haue of veluet other 20 yeardes (to make vp the 40 yeardes) and that woulde coste 400 s. So that the total of the price of the damaske is lesse than the summe paide for veluet 160 s, and should be more by 48. therefore the first errour is 208 too little. Then begin I again, and suppose he boughte of Damaske 30 yeardes that cost 360 s, then had he but 10 yeardes of Veluet, which coste 200 s: and nowe the price of Damaske is greater than the price of the Veluet by 160

shillings, and shoulde be but 48, therefore is the second erroure 112 too muche, whiche I sette in forme of a figure as here doeth ap∣peare. Then doe I 〈 math 〉〈 math 〉 multiplie in crosse wayes. 280 by 30 and the sum wyll be 6240. Also I multiplie 112 by 20, and there wil amounte 2240. And in as muche as the signes of the errours be vnlike, I knowe I muste worke by Addition, therefore adde I those two to∣tals togither, and they make 8480, whiche is the diuidend: then adde I also the two errors togither. 208 and 112, and they make 320, whiche is the diuisor. Wherefore diuiding 8480, by 320, the quotient wil be. 26 ½, whi∣che is the true summe of yeards of Damaske that he boughte: and in Veluet 13 yeardes ½, and that appeared by examination thus: 26 ½ yeardes of Damaske at 12 s. the yeard ma∣keth 318 s, then in Veluet he hadde but 13 yeards and ½. that cost 270 s at 20 s. y^e yeard. Nowe Subtracte 270 out of 318, and there wil remain 48, which is the number of shil∣lings that the Damaske did coste more than the Veluet.

M

Now shal you haue a questiō of an other kinde.* 1.97

There are thrée men that do owe money to me, and I haue forgotten what the total sum is, and what the particulars be.

Scho

Why? then is it impossible to know the debt.

Maister.

Peace ye are to hastie: there is more helpe in it than you yet sée: I haue thrée seuerall notes, whereby it appeareth that I did conferre their debts togither, and founde the debt of the first and the seconde to amount to 47 lb, the debt of the first man and the third did make 71 lb and the second man his debte with the third, did rise to 88 lb. Now can you tell what euerie mā did owe, and what was the whole total?

Scholer.

Naye in good faith: but as I per∣ceiue that it must be founde by coniecture, so wil I gesse at it, supposing that the firste man did owe 20 lb, and the second mā 30, and the third.

M.

Nay stay there for you are to far gone alreadie,* 1.98 you maye not suppose a seueral sum for euerie mā, for it is ynough to suppose one summe for the firste man, and let the other rise as the question importeth. Therefore sée∣ing

you set the first man his debt to be 20 lb, the seconde man can not owe 30 lb, for the de∣claration is that their debtes added togither, did make 47 lb. so must the seconde man hys debte be but 27 lb. Nowe this seconde debte with the thirde muste make 88, therfore sub∣tract 27 out of 88, and there wil remaine 61, as the third man his debt. Then saith the de∣claration, that the first and third mens debts do make 71: but by this suppositiō they make 8•• that is 10 too much: whiche I muste set for the 〈◊〉〈◊〉 error. Now worke you the second po∣sition.

Sc

I suppose the first mans debt to be 24 lb, then must the second mans debt (by your de∣claration) be but 2•• lb. séeing both they make but 47 lb. Also the seconde man his debt wyth the third, doe make 88 lb, and the second man oweth but 2••, therefore the thirde man must d••••e 65 lb. Now the third mans debt with the first, shoulde make by the declaration 71 lb, & they do make 89 lb: that is 18 lb too muche: and that is the seconde errour, which I sette downe with the first, and 〈 math 〉〈 math 〉 their positiō in this forme and then do I multiply in crosse wayes 20 by 18, &

it is 360. Also 10 by 24 maketh 240. And be∣cause the signes of the errors be like, I must worke by subtraction: therfore I subtract 240 out of ••60 and there resteth 120, whiche is the diuidend: then doe I subtract 10 out of 18 by the same reason, and so is the diuisor 8, which is found 15 times in 120, therefore I say that the first man did owe 15 lb, and then the second man muste owe 32 lb, for those 2 doe make 47 lb, and the third man his debt is 56 for so much remaineth if I bate 15 out of 71, or if I take 32 out of 88.

M

For the third exāple take this easie que∣stion for y^e variety in worke.* 1.99 Two mē hauing seueral sums which I know not, do thus talk togither: y^e firste saith to the 2, If you giue me 2 s of your money, thē shal I haue 3 times so much money as you: the 2 answereth: It wer more reason, that our sums wer made equal, and so will it be, if you giue me •• s of youre mony. Now gesse what ech of them had.

Scholer.

I imagine that the firste had 9 s.* 1.100

Maister.

Consider euermore in your ima∣ginations that you take a likelye summe, as in this question take suche a sum that hauing 2 added vnto it, may be diuided into 3 partes euen.

Scholer.

Why? I remember you said be∣fore, it forced not howe fondely so euer I ges∣sed.

M

As for the possibilitie of the solution it is truth, but for easinesse in worke, the aptest numbers are most conuenient.

Scholer.

I thoughte no lesse, and there∣fore I toke 9 as an apte number to be parted into: but I perceiue I shoulde haue conside∣red the aptnesse of that partition after the ad∣dition of 2 vnto it, and then 7 had bene more méeter.

M

That is trueth, and then shoulde the se∣conde man his sum be 5: for although he haue nowe but the third part of 9 that is 3, yet you must remember that he lent the first man 2, and so had he 5.

Scholer

Then to goe forwarde: if the se∣conde man had 3 of the firste mā, then should he haue 8, and the first man but 4, so hath hée double to the first man: yet he said in the que∣stion they should haue equall: wherfore it ap∣peareth that he hath 4 to much. Therefore I note that error with his supposition, and gesse againe that he hath 10 s: whereunto I adde 2 shillings borrowed of the seconde man and then hath he 12 shillings, so the seconde

man hath remayning but foure, wherevnto if I adde the 2 that he lent to the firste man, so had he but 6 s at the beginning. Thē take 3 shillings from the first man, and giue to the seconde, then hath the first man but 7, and the seconde hath 9, whiche are not equall, but there are 2 to manye, 〈 math 〉〈 math 〉 wherfore I set down bothe the positions with their errors as here you sée, and mul∣tiplie a crosse, so commeth there 40 and 14: and bycause the signes be like. I take 14 out of 40, and so resteth 26 to be the diuidend, thē likewise I take 2 out of 4, and there resteth 2 by which I diuide 26, and the quotient wyll be 13, which is the summe that the firste man had. And so appeareth that two being added thereto, the summe will be 15, so hath the se∣cond man nowe but 5, and before he had 7: then take thrée from the first, and put to hys seuen so haue eache of them 10, and that is e∣qual, as the question would.

Maister.

For the fourth example take this question. One man saide to an other:* 1.101 I think you had this yeare two thousand lambes: so had I saide the other: but what with paying

the tith of them, and then thrée seueral los∣ses they are muche abated: for at one time I lost halfe as many as I haue now left: and at an other time the thirde time of so many: and the third time ¼ so many. Now gesse you how manie are left.

Sch.

Bicause here is mention made of cer∣taine parts, I must take a number that may haue all those parts: that is to say, ••/2, 1/••▪ and ••/••, whiche wil be 24, howbeit 12 hath the same partes. Therefore firste I take 12 to be the number that doth remaine, so hath he loste 6, 4, and 3, that is 13: and in the whole 25, but it shoulde be 2000.

Maister

Yée are deceiued yet stil: you haue forgotten the 10 part, whiche must be de••al∣ked, that is 200, so there remaineth but 1800 and now go on againe.

Scho.

Then to finde the error, I take 25 out of 1800, and there remayneth 1775 to fewe, whiche I set for the firste errour. Then for the second position I take 24, whose halfe is 12, the third part 8, & the quarter 6, where by riseth 50, which is too little by 1750, ther∣fore I set downe both the positi∣ons 〈 math 〉〈 math 〉 with their errors thus.

And multiplie in crosse ways 1775-1750-

1775 by 24, whereof commeth 42600. Also I multiplie 1750 by 12, and there ariseth 21000. And bicause the signes are like, I doe subtract the one from the other, and so remai∣neth the diuidend, 21600: then do I subtracte 1750 out of 1775, and there resteth 25: by whiche I diuide 21600, and the quotiente is 864, wherof the halfe is 432, and the thyrde part is 288, the quarter is 216, which 〈 math 〉〈 math 〉 all being added togither, will make 1800. And if you adde thereto the tenth which was abated before, then wil the whole sum be 2000. And nowe doeth there come a question to my memorie which was demaunded of me, but I was not able to answere to it, and now me thinketh I could solue it

M.

Propone your question.

Sch.

There is supposed a Lawe made that (for further tillage) euerie man that doeth kéepe shéepe,* 1.102 shall for euerie ••0 sheepe eare and sowe one acre of grounde: and for his al∣lowance in shéepe pasture, ther is appointes for euerie foure shéepe one acre of pasture Nowe is there a riche sheepemaister 〈◊〉〈◊〉 hathe 7000 acres of grounde, and woulde gladlie kéepe as manye shéepe as he might by

that Statute, I demaunde how many shéepe shal he keepe.

M.

Answere to the question your selfe.

Scho.

First I suppose he maye keepe 500 shéepe, and for them he shall haue in pasture after the rate of 4 shéepe to an acre, 125 acres, and in arable grounde 50 acres, that is 175 in al: but this error is to litle by 6825. Ther∣fore I gesse againe, that he may kéepe 1000 shéepe, that is in pasture 250 acres: and in til∣lage 100 acres, which maketh 350: that is too little by 6650.

〈 math 〉〈 math 〉 These both er∣rours wyth their positions I sette downe as you sée, and multiplie in crosse 6825 by 1000, & it maketh 6825000. Then I multiplie 6650 by 500, and it doeth amounte to 3325000, whiche summe I doe subtract out of the firste and there remaineth 3500000 as the diuidende. Also I doe sub∣tracte the lesser error out of the greater, and so remaineth 175, by which I diuide the said diuidend, and the quotient wil be 20000, so that I sée, that by this rate he y^t hathe 7000 acres of grounde, may kéepe 20000 shéepe:

and thereby I coniecture, that many menne may kéepe so many shéepe, for many men (as the common talke) haue so manye acres of ground.

Maister.

That talke is not likely, for so much ground is in cōpasse aboue 48 ¾ miles, leaue this talke and returne to your questi∣ons, leaste your pointing be scarce well ta∣ken.

Scholer.

Indéede I doe remember, that the Egiptians did grudge so muche againste sheepeheardes till at length they smarted for it, and yet they were but smal shéepemaisters to some men that be now, and the sheepe are waxen so fierce nowe & so mightie, that none can withstand them but the Lion.

Maist.

I perceiue you talke as you heare some other:* 1.103 but to the worke of your questi∣on: both this laste question, and the nexte be∣fore might be wrought without the seconde position, by the rule of proportion, as this. When in this question ye found in the firste errour, that for 500 shéepe, there muste bee 175 acres, then might you reduce it to the Golden rule, thus

If 175 acres wil ad∣mit 〈 math 〉〈 math 〉 in allowance 500

shéepe than 7000 will haue 20000. And so by one position with the helpe of the Golden rule may you answeare that question. Like∣wise for the question of Lambes, when you 〈 math 〉〈 math 〉 had founde that 12 came of 25, you mighte haue set the figure thus as yée see & haue said: If 25 do leaue but 12, what shall 180 leaue? and it woulde appeare to be 864.

Scholer.

Sir, I thanke you for this ayde, for it doeth muche shorten the worke of thys rule.

Maister.

* 1.104Yet againe I will shewe you an other way, to answeare to this laste question without this rule of False position, and that by the rule of Felowshippe, for it appeareth in the proponing of the question, that 10 shéepe must haue in pasture 2 acres and ½, and for them must there be eared but one acre: so it followeth, that for two acres eared, there must be fiue set to pasture. And if you putte them both into one summe, they wil make 7. Therfore looke what proportion 7 being this totall, doth beare to 5 and to 2, such proporti∣on shall any totall in this question beare to the pasture ground, and the eared ground.

Scholer.

This serueth wonderous aptly. Therfore to proue it, I demaund this by the former supposition: If a man haue ••00 a∣cres, how much shall he leaue in pasture, and howe much shal he turne to tillage? You say that as 7 is to 5, so shall 300 be to the acres of pasture: and as 7 is to ••, so is ••00 to the acres of tillage, whereof for both I haue sette examples here following, 〈 math 〉〈 math 〉 whereby appeereth that of pasture there shall be 214 2/7 acres, and of tillage 85 ••/7 which both summes ad∣ded togyther, doe make 300.

Maister.

Now take an other example: A man hath three siluer cuppes with one couer, the couer wayeth 18 ounces the seconde cup weyeth euen halfe the waight of the first and the third. Now if the couer be put to the first cup, they wey iuste as muche as all the thrée cups doe wey: and if the couer be ioined with the seconde cuppe, they wey as muche as the second twice, and the third: and if the couer be put to the thirde cup, they wil make twice as much as the first and the second cup. Now trie you what was the iust waight of euery cup.

Scholer.

I doe set the waight of the firste cup to be 9 ounces: then in as much as these two (that is to say, the couer and the first cup) do wey the waight of the thrée cups, I sée that the three cuppes must wey 27 ounces, for so much is 18 and 9. Also because the firste and the third do wey double so much as the secōd, therefore is it the third parte of that waight, that is 9, and then woulde it followe, that the third cup also should wey 9 ounces, but then the question saith, that the couer being ioined to the second cup, they wey as muche as the second twice, and the third once, that should be 27, and so it doth: then being ioined with the third cup, they should wey twice as much as the first and the second, that should be 36, and they wey but 27, so is that errour 9 too little. Then beginne I againe, and saye, that the firste cup doth wey 12 ounces, whiche I ioine with the couer, and they make 30 oun∣ces: then séeing the second is ⅓ of that waight, it muste néedes wey 10 ounces, and the third muste wey 8 ounces, seeing the first and the third must wey 20 ounces. Now putte I the couer to the seconde cuppe, and they wey 28 ounces, which should be euen so: then ioine I the couer with the thirde Cuppe, and so

should it wey twice the firste, and the second, that is 44 ounces, and they doe wey but 26, that is 18 too little: those 〈 math 〉〈 math 〉 errours with their posi∣tions I set downe, and multiply in crosse ways 9 by 12, whereof com∣meth 108. Also 9 by 18, and that yéeldeth 162: and in as much as the signes be like, I abate the lesser out of the greater, and there doth remaine 54. Then doe I also abate the lesser errour from the greater, and so remai∣neth 9, by which I diuide 54, and the quoti∣ent is 6: which I take for the true waight of the first cuppe: which being ioined with the couer muste wey as much as the thrée cups, so do they wey but 24 ounces. Then séeing the secōd cup is the third part of that weight, for the other two cuppes (you say) muste wey double his weight, the weight of the second cuppe is 8 ounces, and so the waight of the the thirde must be 10 ounces. Nowe put the couer to the seconde cuppe, and it will make 26 ounces: that muste be the waight of the seconde twice, and the thirde once, that is twice s, and once 10, and so is it. Againe, putte the couer to the third cuppe of 10 oun∣ces,

and they must wey twice as much as the firste and the second, that is 28: and so is all agréeable.

Maister.

Then aunsweare to this questi∣on.

* 1.105There is a Cesterne with foure cockes, containing 72 barrels of water: and if the greatest cocke be opened the water wil auoid cleane in sixe houres: at the seconde cocke it wil aske eight houres: at the thirde cocke it will auoide in no lesse than nine houres: and at the smallest it wil require twelue houres. Nowe I demaunde, in what spaes will it a∣uoide, all the cockes beeing set open?

Scholer.

Firste I imagine that it will a∣uoide in two houres.

Maister.

Then muste there auoide by the first cocke 1/•• of the water, that is ••4 barrels, and by the second cocke ¼, that is 18, and by the third cocke 2/•• that is 16 barrelles, and by the smallest cocke ⅙, that is 12 barrelles, all which summes put togither do make 70, as by their addition it doth appéere, but it should be 72, therefore the errour is 2 too fewe.

Scholer.

Then I begin 〈 math 〉〈 math 〉 againe by youre fauour, by∣cause I think I vnderstand the worke, and putte thrée houres for the due time: so shall there runne out at the greatest cocke ½, that is 36 barrels, and at the seconde hole ••/8, that is 27, and at the thirde cocke ••/3 that is 34, and at the smallest hole ¼, that is 18 barrels, which al togither do make 105, and should be but 72, so is it too muche by ••3, therefore do I set the 〈 math 〉〈 math 〉 errors in order of the figure with their positions, and worke by multiplication, in crosse, saying: 2 times 3 is 6 and 2 times 33 maketh 66: and because the signes are vnlike, I must adde those two to∣talles togither, which make 72: also I adde the two errours, and they make 35, by whi∣che I diuide 72, and the quotient riseth 2 2/25, whereby I sée that all the cockes beyng set o∣pen, the water wil auoide in 2 houres, and ••2/35 of an houre.

Maister.

This exercise maketh you to grow expert in the rule. Therefore I wil in∣ure you somewhat more w^t a questiō or two.

There were two men that had bene parte∣ners, and had in accompt betwene them 300 duckets: whereof the one shoulde haue for his parte 180, and the other 120: but in the parting of them they fel at variannce, so that eche of them catched as many as he coulde: yet afterwarde being reconciled, they agréed that he which had gotten most parte of them, shoulde lay downe ¾ of them againe, and hée that had gotten least, shoulde laye downe ⅓ of those which he had taken, and then parting them vnto two equall partes, eache man to haue halfe thereof, and so had they their iuste portions, as they ought: nowe I demaunde of you what eache of them had gotten by the scambling?

Scholer.

I suppose he that had leaste, gotte 108 duckats, then the other had 192: where∣fore in laying downe againe of the 192, there was put downe ¾ that is, 144, and so had he left but 48. Also of the 108: there was layde downe 36, that is ••/••, and so he had lefte 72. Then I put togither 144, and 36, and it ma∣keth 180 which I parte into two partes e∣uen, and so commeth 90 to be giuen to eache of them: which summe put to 72, maketh 162, and ioined to 148, it maketh 238: and now I

doubt how I shal go forwarde.

Maister.

You néede not to take but one of them which you list, the greater or the smal∣ler, for all commeth to one purpose: and so may you compare it that you take to anye of the other summes, remembring that you make comparison to the same in the seconde worke: as for example of the firste parte, If you compare 138 with the lesser summe due, that is 120, so is it 18 too muche: and if you compare it with the greater summe, then is it 42 too little. Againe, if you compare 162 to the greater summe, the errour will be 18 as it was in the other: but it wil haue a con∣trarie signe: and if you compare it with the lesser summe it will be 42 too much: so that the errour both wayes is eyther 18 or 42: & as for the signes it little forceth, for in them is nothing considered here, but likenesse and vnlikenesse, which in this case, doth neyther further nor hinder. But now go on with the worke.

Scholer.

If it be so, then am I out of my greatest doubt Then I ioine that 90 (which I founde as the halfe of the latter partition) vnto 48, which is left with the one man, and so hath he 138, whiche (I may say) is 18 too ma∣ny

for the least should be but 120: that error doe I note, and then make a newe position, supposing the one man to haue 204, and the other to haue 96, wherefore of the 204 there must be laide downe 153, and so remaineth with him 51. Also of the 96 there muste bée laid downe ⅛, that is 32, and so resteth with that man. 64. Nowe of the 153 and 32 I make one summe as 185, whiche I muste diuide into 2 equall partes. and so eche man shall haue 92 ½, wherevnto if I adde their former portions reserued, then the one shall haue 156 ½ and the other hath 145 ½. Wher∣fore I take the lesser summe nowe agayne, as I did before, that is 143 ½, and finde that he hath too many by 23 ½, for he shoulde haue but 120, so haue I for my two positions two errors, which I set down, 〈 math 〉〈 math 〉 as here may be seene, eche errour vnder his position, and then by the rule I doe multiplye in crosse wayes 108 by 23 ½ and there riseth 2538 whiche I note then againe I multiplye 96 by 18, and thereof amounteth 1728. Now because the signes are both like, that is bothe too ma∣ny, I must work be Subtraction, and so aba∣ting

1728 out of 2538, there wil reste for the diuidend 810: then for the diuisor I subtract 18 out of 23 ½ and there remaineth 5 ½, by whiche I diuide 810, and the quotient will be 147 ••/33••, which is the iust portion of him that had the least summe. And if I doe sub∣tracte it out of 300 being the totall summe, then wil there remaine 152 9/1••, as the portion that the other did get.

Maister.

For the proofe of this worke, you may choose whether you will examine those numbers according to the forme of the que∣stion, or else worke by other two positions for to finde the second number: and if those positions bring the same numbers that didde amount by the first two positions, then dothe eche worke confirme other.

Scholer.

By your patience, I will proue both wayes, not only to see their agréement, but also to accustome my mind to those wor∣kes: for I perceiue it is exercise that must bée the chiefe engrauer of these rules in my me∣morie.

Maister.

You consider it wel: then goe to.

Scholer.

Firste I will by two other po∣sitions trie to finde the portion of him which

had moste.

Maister.

Although you may doe it with a∣ny positions, yet to sée the agréement of your work the better, take the same positions that you did before, comparing them nowe to the greater, as you did before vnto the lesser.

Scholer.

Then I suppose, that he that had moste, had 192, so had the other 108. Nowe if I take ¼ of 192, that wil be 144, and there will reste to that man but 48. And from the second which had 108, if I take ⅓, that is 36, there wil remaine to him 72: then ioyning 144 with 36, it will make 180, the halfe whereof being 90. If I adde to eche of those two mens portions remaining with them, the one shal haue 138, and the other 162, of which two I take the greater (that is 162) and sée it to be 18 too fewe, for it shoulde bée 180, that errour I note vnder his position. Then for the seconde position I take (as I did before) 204 for the one, and so resteth 96 for the other: then take I ¾ of ••04 and it wil be 153, and there resteth to him 51. Also of the 96 I take ¼ that is 32, and there remay∣neth to him 64. Now put I that 32 to 15••, and it yéeldeth 105: whiche being parted in equall valewes, maketh 92 ½. to be added to

eche mans remainder, and so the one hathe 143 ½, and the other 156 ½: wherefore I take the greatest summe, and it is 23 ½ too lit∣tle, that doe I note also, and sette both these errours vnder their positions, as in this ex∣ample folowing doth appeare.

And then multiplying 〈 math 〉〈 math 〉 192 by 23 ½, there doth a∣rise 4512.

Again, I multiply 204 by 18, & it maketh 3672, whiche I doe subtracte out of 4512, because the signes be like, & there resteth 840 for the diuidend: then subtracting 18 out of 23 ½, there will remaine 5 ½, which I must take for the Diuisor. And so diuiding 840 by 5 ½, the quotient wil be 152 8/11, whereby I haue found an agréeable summe to that whiche I found by the former positions, for hym that had moste, whiche if I doe subtracte out of 300, that is the totall, there wil rest 147 ••/11, which was the portion of him that hadde the least parte.

Maister.

So by diuers positions you sée, that one doeth confirme the worke of the o∣ther. Nowe examine those two numbers by the forme of the question, and so shall you

proue your worke good also.

Scholer.

If that he whiche gate most, had 152 3/11, then muste he laye downe ¾ of hys summe, that is 114 6/11▪ and so shall remaine with him, but onely 38 2/11 The other which had leaste, that is 147 3/11. muste put downe of his sum ½. that is 49 1/11, and so doth there remaine with him yet 98 3/1••. Then doe I adde togither 114 6/11 and 49 1/11, and it wil make 163 ••/1••, whiche I muste part into two equall partes, and that will be 81 9/11 to bee giuen to eche of them: so putting 81 9/11 vnto 38 2/11, there doth amount 120 iuste, whyche is the true portion of him that shoulde haue the lesser summe: and adding 81 9/11 to 98 2/11, the totall will be 180. the true portion of the other. And so is the worke by this proofe al∣so tried to be good. And this I marke by the way, that in their scambling, hée gate moste (as it chanceth often) that ought to haue had least by iust partition.

Maister.

Let your study be to learne truth and iuste arte of Proportion, and to distri∣bute and parte according therevnto, as often as occasiō shal be ministred. And here would I make an ende of this rule, saue that I re∣member

one pleasant question whiche I can not ouerpasse, which I wil declare somwhat largely, because you shal as well vnderstand some reason in the pleasaunt inuention, as apte procéeding in the wittie working there∣of.

Hiero King of the Syracusanes in Sicilia,* 1.106 hadde caused to bée made a Crowne of Golde of a wonderfull waight, to be offered for hys good successe in warres: in making whereof, the Goldsmith fraudulently tooke out a cer∣taine portion of Golde, and putte in siluer for it, so that there was nothing abated of the full weight, although there was much of the valewe diminished. Which thing at length being vttered, (as no euill can alwayes lye hidde) the King was sore moued, and beyng desirous to trie the truth without breaking of the Crowne, proponed the doubt to Archi∣mides, vnto whose wit nothing séemed vn∣possible, whiche althoughe presentlye hee coulde not aunsweare vnto, yet hée had good hope to deuise some pollicie for that inuenti∣on. And so musing thereon, as he chanced to enter into a baine full of water to washe him, he obserued that as his body entred into the baine, the water did runne ouer the tub:

whereby his ready wit of suche small effectes coniecturing greater workes, conceyued by and by a reason of solution to the Kings que∣stion, & therefore reioicing excéedingly more than if he had gotten the Crowne it selfe, for∣gate that he was naked, and so ranne home, crying as he ran, 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, I haue founde, I haue founde. And there vppon caused twoo massie péeces, one of golde, and an other of siluer to be prepared of the same weight that the saide Crowne was of: and considering that golde is heauier of nature than siluer, and therfore golde of like waight with siluer, must néedes occupie lesse roume, by reason it is more compact & sound in substance, he was assured, that putting the masse of golde into a vessel brimme full of water, there would not so much water run ouer, as when he shoulde putte in the siluer masse of the like weight. Wherefore he tried both, & noted not onelye the quantities of the water at eche time, but also the difference or excesse of the one aboue the other, wherby he learned what proportiō in quantitie is betwéene gold and siluer of e∣quall waight. And then putting the crown it self into the vessell of water brimme full (as before) marked how much water did run out

then, & comparing it with the water that rā out when the gold was put in, noted howe much it did excéede that: & likewise compa∣ring it to the water that rā out of the siluer, marked how much it was lesse than that: & by those proportions found out the iust quā∣titie of gold y^t was taken out of the crown, & how much siluer was put in stéede of it. But séeing Vitruuius whiche writeth this histo∣rie, doth not declare the particular worke of this trial, it shal be no inconuenience to sup∣pose an example for declarations sake, wher∣in although the true and iust proportions be not expressed, yet the forme of triall shall be truely set forth And for an example, I sup∣pose the weighte of the Crowne to be 8 lb, and so of each of the other two Masses. And when the masse of Golde was put into the water, I imagine that there ran out 2 poūd of water: and when the masse of siluer was put in, I suppose there ran out 3. pound ½ A∣gaine when the crowne was put in, there rā out 2 pound ¼ Now to know what quanti∣tie of siluer was in the Crowne, worke by the rule of false position, and imagine that there was 2 pound of siluer: then must there be 6 pounde of Golde▪ Then say thus

by the rule of Proportion: If 8 pound of gold doe expell 2 lb of water, what shal 6 lb. expell? and it wil be 1 pound ½. Againe for the siluer: If 8 lb of siluer expell •• lb ½ of water, what shall 2 lb of siluer put out? it will be ⅞. Nowe adde those two weightes of water togither, and they will make 2 lb 3/•• and it shoulde be by the supposition 2 lb ••/••, so is it too muche by ⅛

Sch▪

Nowe doe I vnderstande the worke as I thinke, therfore I pray you let me work the rest of the question. And bycause this first supposition did erre, I note that positiō, and his errour, and take a newe position, estée∣ming the siluer to be but one pound, so muste there be in Golde 7 pounde, Then say I: If 8 lb of Gold yéeld 2 lb of water, what shall 7 lb yéelde? and it will be 1 lb. ¾. Againe if 8 lb, of siluer expel 3 lb. ½ of water what shall 1 lb ex∣pell? and it wil be ••/2 7/6. Nowe muste I adde those two sums togither, and they make 2 lb, ••/16 and they sh••ulde make 2 lb ¼ so is it too lit∣tle by 2/16 Therfore I set y^e positiōs with their errours in order, as here 〈 math 〉〈 math 〉 foloweth. And thē I mul∣tiplie in crossewaies 2 by 6/11 and it maketh ⅛: like∣wise 1 multiplied by ⅛:

maketh 2/8. And because the signes be vnlike, I must adde those two summe, whiche make ¼ and that is the diuidend. Again I must adde ⅛ to 1/16, and it wil be ••/••••▪ that is the diuisour. Now I shall diuide 1/•• by ••/•••• and the quotient wil be ••••/•••• that is, 1 ½, whereby I knowe that there was put 1 lb and 1/•• of siluer into the Crowne, and so muche Golde taken out for it.

M

Proue it now by examinatiō according to the question.

Scho.

If there were 1 pounde ••/3 of Siluer then was there of Golde 6 pounde ⅔. Nowe say I by the rule of proporti∣on: 〈 math 〉〈 math 〉 if eighte pound of Golde expell two pounde of water, what shal 6 pound ⅔ expel?

〈 math 〉〈 math 〉 It wil be one pounde ⅔. Againe, if 8 lb of Syluer expell 3 lb ½ of water, what shal 1 ½ expel? It wil be ••/1 7/2.

Now must I ad togither 1 lb ½. and ••/1 7/2 and they wil make 2 lb, 9/••6, that is 2 lb ¼, according to the supposition of the question, whereby I perceiue the worke to be wel done. And as I can not but muche reioyce of thys excel∣lent

inuentiō, so my desire is kindled vehe∣mently to be perfectlye instructed in euerie part therof, and namelie in this point, whe∣ther the portion betwéene water and golde be such, that for 8 lb of gold into a vessel full of water, there shall run out 2 lb of water: & for as much siluer, whether 3 lb ½ of water would auoide?

Ma.

I perceiue your meaning, and conie∣cture your imagination to be thus: that if you knewe the exacte proportion betwéene Gold and Siluer and Water both in theyr waight and quantities, then coulde you ea∣silie finde out the mixtures of them, whiche thing I haue reserued for an other worke that intreateth such matters speciallie. And at this time you muste consider, that you learne Arithmetike, which intreateth of the maner to solue doubtful questions touching number, without regarde what matter is signifyed by that number, else were it ne∣cessarie in Arithmetike to teach all arts, sée∣ing in it may be moued questions of al arts. But seeing you are so desirous to know this thing, I wil tell it you in suche a sorte, that you shall practise your arte in finding it, and propounde it in forme of a question. Gold

beareth greater proportion to water than sil∣uer doth, and their two proportiōs be in pro∣portion togither as 4••/25. But to help you some∣what in this riddle, you shall note that the proportion of quicke siluer vnto water, is the iust middle number proportionall in Pro∣gression Geometricall, betweene the propor∣tions of Gold and siluer vnto water. And his proportion is as 29••/••1. Now if you wil know the iust numbers of these 3 proportions then must you finde out 3 numbers in Progressi∣on geometrical, wherof the middlemost must be 290/21, and the first must be vnto the last, as 25 to 48. And thus I will leaue you to finde those numbers when you be at leasure.

Scholer.

Yet sir I thanke you heartilie for this muche, for nowe I sée the possibilitie to finde them out. Howbeit, bycause this questi∣on séemeth straunge, if it might please you to instructe me somewhat in the order of wor∣king for it, I should the more easilie finde the true working.

M.

You desire too much ease if you wil stu∣die for nothing: therefore to occasion you to studie y^e better, I will leaue this doubt wholy to your own search. But as touching the ge∣neraltie of the rule, Archimedes néeded not

to take two masses of golde and siluer equall in waight with the crowne, for the proporti∣on might as wel be foūd in any other waight yea althoughe the masse of golde were of one waight and the masse of siluer of another. As for example. If the crowne were of 8 pounde waighte, as I did suppose, and I haue not so muche other fine golde, but onely 1 lb, and trying that by water, & finding that it doth expell but ••/4 of an ounce of water, yet then by it may I inferre, that 8 pound of gold would expell 6 ounces of water And likewise of the siluer: wherof if I had but 2 pound, and finde that it doth expel thrée ounces of water, then might I affirme that 8 pound woulde expell 12 ounces, that is 1 lb waight. And so is it, as good as if the 3 masses were al of one weight. And thus for this time I wil make an end of this other part of Arithmetike.

Sc.

Although I can not sufficientlie thanke you for this, yet your promise made me to loke for the art of extraction of rootes, wherof hitherto I haue learned nothing.

Maister.

I wil not breake my promise, but intend (God willing) to performe it within these thrée or foure monethes, if I perceyue this my paines to be well taken in the mean

season. And you shall not repent the tarrying for it: for it shal be increased by the tarrying. And in the meane time, you shall take thys Addition, not for the second part of Arithme∣tike which I promised, but for an augmenta∣tion of the firste parte, vnto which I woulde haue annexed the extraction of Rootes square and cubike, namely for examples of the Sta∣tute of Assise of wood, but that in the seconde parte I muste write of diuers other rootes, and thought it beste to reserue those rules al∣so with their examples vnto the same seconde parte.

Scholer.

Sir, althoughe I can not recom∣pence your goodnesse, yet I shall alwayes do mine endeuour to occasion you not to repent your benefite on me thus employed.

Maister.

That recompence is sufficient for your parte.

FINIS.

Notes

• * 1.1

  Addition of 2 summes.
• * 1.2

  To ad 8 summes to∣gither.
• * 1.3

  A prooef.
• * 1.4

  ••nother ••ay of ad∣••••tion.
• * 1.5

  ☞
• * 1.6

  ☜
• * 1.7

  ☜
• * 1.8

  An other forme of Multiplica∣tion.
• * 1.9

  The summe of the daies sith Christs incarnation
• * 1.10

  ☜
• * 1.11

  Example of vvages
• * 1.12

  Example of purchase.
• * 1.13

  ☞
• * 1.14

  Auditours accompt.
• * 1.15

  ☜
• * 1.16

  VVhat a Fraction is.
• * 1.17

  VVhat a Fraction in properlie.
• * 1.18

  ☜
• * 1.19

  Numera∣tor, and Denomi∣nator.
• * 1.20

  ☜
• * 1.21

  ☞
• * 1.22

  Three se∣uerall va¦rieties.
• * 1.23

  ☞
• * 1.24

  Note hovve 〈…〉〈…〉 2 numbers.
• * 1.25

  ☞
• * 1.26

  Note.
• * 1.27

  Fiue varie∣ties of Re∣duction.
• * 1.28

  The first sort of Re∣duction.
• * 1.29

  Hovv to reduce fra∣ctions of diuerse de∣nominati∣ons into one Deno∣mination.
• * 1.30

  Note the Reduction of three Fractions or more to one.
• * 1.31

  The first varietie of this Redu∣ction.
• * 1.32

  The second varietie.
• * 1.33

  The thirde ••arietie.
• * 1.34

  Proofe.
• * 1.35

  ☜
• * 1.36

  Reductiō of fractions of fractions into one fraction and deno∣mination.
• * 1.37

  The third forme of Reduction.
• * 1.38

  Reductiō of improper fri¦ctions into vnits vvith their proper Fractions.
• * 1.39

  Reductiō of vvhole num∣bers, either alone, or ioi∣ned vvith fractiōs into ••ractions.
• * 1.40

  ☜
• * 1.41

  The fourth kind of Re∣duction.
• * 1.42

  Tearms of Fractions.
• * 1.43

  Reduction of Fracti∣ons into their smal∣lest tearms.
• * 1.44

  An other vvay for to vvorke Reduction.
• * 1.45

  ☜
• * 1.46

  The fift kind of Reduction
• * 1.47

  Reduction of vvhole numbers into Fractions.
• * 1.48

  ••vv to mul¦••ie mixt ••he••s.
• * 1.49

  〈 math 〉〈 math 〉
• * 1.50

  ••ision to ••de a ••ole nū∣•• by a ••ion.
• * 1.51

  ••iuide ••ction vvhole ••er.
• * 1.52

  Another briefe vvay
• * 1.53

  〈◊〉〈◊〉 adde ••••ctions of ••e deno∣••••nation.
• * 1.54

  ••o ad fra∣••••••ons of d¦••rse deno∣••••nations.
• * 1.55

  ••eatest •• fra∣••
• * 1.56

  A question of Yeluet.
• * 1.57

  A question of siluer.
• * 1.58

  The proofe of the Gol∣den rule.
• * 1.59

  The Backer rule.
• * 1.60

  A question of lone.
• * 1.61

  Statute of Assise of breade and ale.
• * 1.62

  Question.
• * 1.63

  A pounde vveight.
• * 1.64

  A question of measure o•• ground▪
• * 1.65

  A statute measures.
• * 1.66

  An Acre.
• * 1.67

  Note this ••rour.
• * 1.68

  ☞
• * 1.69

  A question of vnequal societie.
• * 1.70

  ☞
• * 1.71

  The reason of this rule.
• * 1.72

  The proofe by Additiō
• * 1.73

  The iust proofe.
• * 1.74

  A question of building
• * 1.75

  An impos∣sible que∣stion.
• * 1.76

  ☜
• * 1.77

  The rule.
• * 1.78

  A question of a testament.
• * 1.79

  To finde three nūbers in any pro∣portion.
• * 1.80

  To finde the proportion betvvene 2. numbers.
• * 1.81

  An other question of •• Testament.
• * 1.82

  The rule of mixture.
• * 1.83

  The reas•••• of this ru••
• * 1.84

  ☞
• * 1.85

  A question of mixing of vvine.
• * 1.86

  The com∣mon price. ☞
• * 1.87

  The proo••e of 〈…〉〈…〉.
• * 1.88

  The variati∣on of this question.
• * 1.89

  A question ••f spices.
• * 1.90

  The occasi∣on of the name.
• * 1.91

  ☞
• * 1.92

  ☜
• * 1.93

  Note:
• * 1.94

  The profe of this rule.
• * 1.95

  ☜
• * 1.96

  A question of vvares.
• * 1.97

  A questio•• of debt.
• * 1.98

  ☜
• * 1.99

  The third question.
• * 1.100

  Note:
• * 1.101

  The fourth example.
• * 1.102

  A question•• Of sheepe and tillage.
• * 1.103

  An other vvay of vvorking.
• * 1.104

  An other vvay yet.
• * 1.105

  A question of vvater.
• * 1.106

  An example of mixture of Gold and Siluer.