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 GOLDEN Rule.

Maister.

BY order of the science (as men haue taughte it) there should fellow nexte the extraction of Rootes of number, whiche because it is somewhat harde for you, yet I will let it passe for a while, and will teache you the feate of the rule of Proportions, whiche for his ex∣cellencie is called the Golden rule. Whose vse is, by thrée numbers knowen, to finde out an other vnknowen, whiche you desire to knowe: as thus. If you pay for your boorde for thrée moneths 16 shillings, howe muche shal you pay for 8 moneths.

To know this and all such like questions, you shall consider which two of your 3 num∣bers be of one denominatiō, and set those two the one ouer the other, so that the vndermoste

be it that the question is asked of: as in my question 3 and 8 be both of one denominati∣on, for they both be monethes, and because 8 is the number that the question is asked of, I set them one ouer the other, 〈 math 〉〈 math 〉 and 8 vndermoste, thus, with suche a crooked draught of lines. Then doe I set the other number whiche is 16, a∣gainste 3, at the right side of 〈 math 〉〈 math 〉 the line, thus.

And nowe to knowe my question, thys must I doe: I muste multiplie the lowermost on the left side, by that on the right side, and the summe that amounteth I muste diuide by the highest, on the left side. Or in playner wordes thus: I shall multiplie the number of whiche the question is asked (whyche is called the Thirde number) by the number of an other denomination,* 1.1 (whiche is cal∣led the Seconde) and that summe that amoū∣teth muste I diuide by the summe of lyke denomination, whiche is called the Firste. Then for the knowledge of this question, I multiplie 8 into 16, and there amoūteth 128, whiche I diuide by 3, and it yéeldeth 42 shil∣lings, and 2 s remaineth, whiche I turne into pennies, and they be 24 d, of whiche

the third part is 8 d, so the third part of 128 s is 42 s, 8 d: which summe I write at the right hande of the figure a∣gainst 〈 math 〉〈 math 〉 8, thus.

Hereby I knowe, that if thée monethes boording doe come to 16 s, that 8 monethes bording wil come to 42 s, 8 d: and likewise of any other like question.

But here must you marke, that the firste number and the thirde be of one denominati∣on, and also the seconde and the fourth, the whiche you séeke: or else be of suche denomi∣nations, that you in working may bring thē into one. As if a man shoulde aske me thys question.

*Twelue wéekes iournying coste me four∣téene French Crownes at 6 s. the péece, how many poundes is that in one yeare. Here you sée no two nūbers of one denomination, But yet in working, you may turne them in∣to like denominations, as thus: turne the one yeare into 52 weekes, and the fourth summe wil be French Crownes, by the order of the working: Then to knowe this question, multiplie the thirde summe 52 by the second 14: and the summe will be 728: that diuide by your firste number 12: and the quotiente

will be 60. Crownes: And 8 Crownes re∣mayning: whiche if you turne into shillings they will be 48 shillings which if you doe di∣uide by your firste number 12 the quotiente will be 4: whiche signifyeth 4 s. put these 60. French Crownes (which make 18 poundes) with the 4 shillings: for 〈 math 〉〈 math 〉 the summe that answe∣reth to the question: And it is the iust expē∣ces of a yeare: And the summe wil be thus.

And take this euermore for a general Rule touching this whole Arte, That the doubtful or vnknowen number, that you woulde be resolued of, shall alwayes be set in the thirde place, note also the first number and the third, must euer be of one nature and denominati∣on, or else must in working be brought to like denomination and then of necessitie must the other number be in the second place.

Remember also, that the place of the firste number is the highest on the left side: and the place of the seconde right against it on y^e right side: the place of the thirde number is vnder the firste, as by those examples you haue séene.

Sc.

This I trust I can do.

Mayster.

But and if the question be asked thus: In 8 wéekes I spend 40. s howe long wil 105 shillings serue me? Here you sée that 8 wéekes aunsweres himselfe, and saieth 40 shillings. But how long time 105 shillings wil serue, you know not. Therefore you shal set 105 in the thirde place, according as I tolde you euen now. And the first place must alwayes be of the same nature or Denomi∣nation, that the third is of, which here is 40. Then must 8 néeds be that other. Now mul∣tiply 105 by 8 and it will be 840 which if you diuide by 40, it will yéelde 21, whiche is the Fourth number, and sheweth howe manye wéekes 105 s. will serue, if you spende 40 s. in eight wéekes.

The figure of thys que∣stion 〈 math 〉〈 math 〉 is this: as if you should saye: If 40 s. serue for 8 wéekes, 105 serue for 21 wéekes.

Other diuersities there be of working by this rule, but I hadde rather that you woulde learne this one well, than at the beginning to trouble your minde with many formes of working, sith this way can doe as much as al

the other, and hereafter you shall learne the o∣ther more conuenientlie.

And for your further aide and instruction to make you better acquainted wyth thys Goldon Rule, I haue here proponed 6 que∣stions, and their aunsweres, whiche I thinke moste conueniente and méete to preferre the desirous to perfecte vnderstanding. The firste foure are all braunches of one Question sproong out of the beste trée, (for a young learner to taste of) that groweth in this Grounde of Artes, for that no manner of Question in the Rule of 3 what so euer it bée, can be proponed, but it muste be com∣prehended, vnder the reason or style of one of these foure.

The Questions be these.

If 15 elles of Cloth coste 7 lb. 10 s: what comes, 27 elles too at that price: Aunsweare. 13 lb. 10 s.

If 27 elles coste 13 lb. 10 s: what are 15

elles worth. Answere. 7 lb. 10 s.

If 27 elles coste 13 lb. 10 s: howe manye elles shal I haue for 7 lb. 10 s. Answere 15 elles.

If I sell 15 elles for 7 lb. 10 s howe ma∣nye elles are to be deliuered for 13 lb. 10 s. Answere. 27 elles,

If 8 pound of any thing cost 16 s. 6 d. what money is to be receiued for 49 pounde: An∣swere. 2 lb 4 s. 11 d.

If 4 lb. of anye thing coste 17 d: what mo∣ney wil 8765 pound of that commoditie cost. Answere. 155 lb. 4 s. 3 d. farthing.

Of all which questions I omitte the work of purpose, you shoulde whet your wit there by at conuenient leasure, to climb ech branch and gather the fruite of them: And doe minde nowe, before we make ad ende of this Rule, to giue you some Instructions of the Bac∣ker Rule of 3. whose order is quite contra∣rye to thys that you haue learned. For in thys Rule hitherto euermore looke how muche the thirde number is greater than the firste, so muche the fourthe number is greater then the seconde. And contrarye wayes: loke howe much the firste summe is greater then the thirde, (if it doe chaunce so)

so much is the second summe greater then the fourth. But in this Rule, there is a con∣trary order, as this: That the greater the third summe is aboue the first, the lesser the fourth sum is beneth the seconde: and this rule therfore you may call the Backer rule,* 1.2 as in example.

If I haue boughte 20 yardes of cloth,* 1.3 of 2 yards breadth, and woulde buy canuas of 3 yards brode to line it withall, howe manye yards should I néede?

Sc.

Why, there is none so broade.

Mai.

I doe not care for that, I doe put this example only for your easie vnderstan∣ding: For if I should put the example in o∣ther measures, it would be harder to vnder∣stand. But nowe to the matter: If you woulde know this question, set your num∣bers as you did before: but you shall mul∣tiplie now the first number by the seconde, and that ariseth thereof, you shall diuide by the third: which thing if you doe here, I meane if you multiplie 30 by 2, it will be 60: which summe if you diuide by 3, there will appeare 20: whereby I knowe, that if 30 yardes of cloth of two yardes broade, should be lined with canuas of thrée yardes

broade, 20 yards of can∣uas 〈 math 〉〈 math 〉 woulde suffice, as this figure sheweth.

And nowe because yée found fault at my exam∣ple, how say you, perceaue you this?

Sc.

Yes sir. I suppose.

Mai.

Then aunswere me to this questiō: how many elles of canuas of elle breadth, will serue to line 20 yards of Saye, of thrée quarters brode.

Sc.

In good faith sir, I cannot tell, for I know not how to bring the summes to like denominations.

Maister.

Then wil I tel you: sith there is mention here of quarters, and againe e∣uerye one of the measures both elles and yardes may be parted into quarters, do you parte them so both in the breadth and length, and then put forth the question by quarters.

Scholer.

Then I shall say thus. Howe manye quarters of canuas of fiue quarters broade, wil line 80 quarters of 3 quarters brode.

Mayster.

Now aunswere to the questi∣on.

Sc.

First I wil set them 〈 math 〉〈 math 〉 downe in their forme thus, for 5 is ioyned with y^e que∣stion, and is therefore the third number: then is 3 the number of the same denomination, I meane because they be both referred to breadth. Now I multi∣plie 80 by 3, and it is 240, which I diuide by 5, and it yieldeth 48. Then saye I, that 48 quarters of 5 quarters broad, will suf∣fice to line 80 quarters of 3 quarters brode.

Ma.

Turne the quarters againe into elles and yardes.

Sch.

Then I say, that 9 elles and thrée quarters of a yarde of elle 〈 math 〉〈 math 〉 broade will serue to lyne 20 yardes of thrée quar∣ters brode, as this figure sheweth.

* M.

Now what say you to this questiō: I lent my friend 400 lb for 7 months, now how much money ought he to lēd me again for 12 moneths, to recompence my curtesie shewed him. Can you aun∣swere 〈 math 〉〈 math 〉 to this.

Sc.

Yes sir I suppose, for I wil set down my nūbers thus

where I multiplie 7 into 400, and it ma∣keth 2800, whiche I diuide by 12, and it yieldeth 233 lb and there is 4 lb. remaining of my Diuision, what shall I doe there∣with.

Mai.

Turne that same 4 lb into s and then diuide it by 12 as you did before.

Sc.

Well sir it shall be done, so haue I 6 s. for my quotient, and yet remaineth 8 s. vpon my diuision.

Ma.

You must also reduce 8 s. into pence, which maketh 96, and diuide that al∣so by your first diuisor.

Sc.

So haue I done, and I finde 8 pence for my quotient and nothing is left.

Ma.

This must you alwayes doe, when any thing remaineth vppon your diuision: whether it be mony, weight, measure, or a∣ny kind of thing whatsoeuer. This rule is so profitable for all estates of men, that for this rule onlie (if there were no more but it) all men were bound highly to estéeme A∣rithmetike.

By this Rule maye a Captaine in warre worke many things, as Maister Digges in his Stratiaticos doth notably declare: Onlie nowe in this my simple addition, for a taste

and incouragemēt I wil enlarge y^e Author with a questiō or ij. more, wishing you, & e∣uerie my coūtriemen or Gentlemē, whatso∣euer, that by nature be anye thing giuen to Millitarie affaires, to be familier and wel acquainted with this Exceliente Arte, the whiche he shall finde not onely at the Sea, but also in the Campe and Fielde seruices, aboundantly to aide him, either in fortifica∣tiō, or in paying of Souldiors wages, how differente soeuer their paye be: Charges of Ordinaunce pouder, shot, Munition and in∣strumentes, whatsoeuer, but now to the question.

If it should chaunce a Captaine whiche hath 40000 souldiers,* 1.4 to be so inclosed with his enemie that he could haue no fresh pur∣ueiance of vittailes, and that the vittailes which he hath, would serue that armie but only 3 moneths, how many men should hée dimisse, to make the vittaile to suffise y^e re∣sidue, 8 moneths? 〈 math 〉〈 math 〉

Sc.

As you taught me, I set the nūbers thus, saying: If 3 moneths suffise 40000, to howe manye will 8 mo∣neths suffice?

To know this I multiplie the first num∣ber 3 into seconde 40000, & it yieldeth 120000, which sum I diuide by 8, & there wil be in the quotiēt 15000, which if I doe subtract from 40000, the 〈 math 〉〈 math 〉 remainer will declare that hée muste dismisse 15000 as this figure sheweth

* Maist.

Nowe aunswere me to this question: If 136 Masons in a moneth bée able to builde a Forte, to preserue the Souldiers from the enemie: And suche expedition requireth that I woulde haue the same finished in eighte dayes, how ma∣nye workemen saye you is there to bée ap∣pointed.

Sch.

As you taught me I 〈 math 〉〈 math 〉 set the numbers thus, saying: If 28 daies require 136 Ma∣sons, what number of men by proportion will 8 dayes bring forth.

To know this I multiplie the first nūber 28 into 236: And it yieldeth me 2808: whiche I diuide by 8. And my quotiente is 476, which is the iust number of Masons that shall supplie this worke. And now me think these questions are very easie.

Ma.

Truly if you take delectation herein you shall finde this Art not onely easie, but wonderfull pleasant & profitable: Now an∣swere me this question, & so wil I make an end of this rule, in whole nūbers hasting y^e sooner to broken nūbers. For had you y^t vn∣derstanding of thē perfectly, not only in this Rule, but in all other: the question in sight might haue ben 10 times more harder to absolue, & yet as easilie & as soone wroughte as this.

Sc.

Your words doth greatly encorage me to be studious to attaine whole numbers, which me think are wōderfull. But might I once attain to be a practicioner in brokē, I should think my self a happy lad.

M.

Now what say you to this, If •••• car∣penters in 2 dayes can make ••00 Staues: estéeming they work but 12 howers a day: And such néede requireth y^t 384 carpenters are set to the finishing of these 200 staues, in what time say you wil they make thē vp

Scholer.

I see here that I muste turne my •• daies into howers, And 〈 math 〉〈 math 〉 so doing I sette my numbers thus

Saying, if 48 men are

24 houres 384 men will make an ende quickely. For it is grounded vpon an olde Prouerbe, many hands make quick spéede.

I multiplie 48 into 24, and it amounteth to 1152, whiche I diuide by 384: and my quotient is 3 houres which is my desire.

* 1.5I take this for a note worthie the mar∣king either in the Rule of thrée, forwarde, or backwarde, when the two numbers art multiplied togither, the Producte is of the same nature, and denomination that the se∣cond number is of.

Maister.

Well, sithe you perceiue nowe the vse of this Rule,* 1.6 I will shewe other which ensue of the same, & firste the double Rule, which is so called, because there is in it double working, by which thing onely it differeth from this.

Sch.

Then by an example I shal vnder∣stand it well ynough.

Ma.

* 1.7So shall you, and let this be the ex∣ample: If the cariage of 100 pound weight 30 miles, doe cost 12 d. how much will the cariage of 500 weight cost, being caryed 100 miles?

Scholer.

I pray you shewe me the wor∣king of it.

M.

You must make 2 workings of it: the first thus. If 100 pound weighte cost 12 d. how much will 500 lb. cost?

Set your figure thus. 〈 math 〉〈 math 〉 And multiplie 500 by 12 and therof amoūteth 6000, which if you diuide by 100, the quotiēt wil be 60, that is the price of 500 for 30 miles.

Thē begin the second worke, saying: if 30 miles cost 60 d, how much wil 100 miles cost? Set your figure thus. 〈 math 〉〈 math 〉

Than multiplie a 100 by 60, wherof amoūteth 6000 which being diuided by 30, will yield 200 d. Than you may say, that so many penies shall cost the cariage of 500 pound waighte 100 miles, after the rate of 12 pence for the 100, caried 30 miles.

Scholer.

Now I perceiue it also.

Mai.

These and such other like questiōs, are to be aunswered much quicker, at one working by the Rule of 3 composed of fiue numbers, whiche here I will not trouble you withall. But at the ende of this Rule will shewe you the worke thereof: not on∣lye of this and the nexte question, but also I will there deliuer thrée or 4 other exam∣ples,

wishing you then to make a com∣parison the one with the other: And so to vse which way you thinke good.

Sc.

Sir I thanke you much for your cur∣tesie, and I long now til this rule be ended, that I shall sée howe I maye behaue my selfe with that newe Rule of 5 numbers, for that I haue euer since you taughte mée hetherto, in the Golden Rule both forward and backward wroughte but with 3 num∣bers only.

Ma.

* 1.8Till we haue done with this, lette vs go on forward: and answere me to this question: 30 bushels of wheat sowed, yiel∣ded in one yeare 360: how manye will 80 bushels yield in 7 yeare.

Sc.

First I saye, that if 30 bushels will yield 360 in 1 yeare, then 80 bushels will yield 960 in 1 yere. Then for the seconde worke I say: If one yeare yield 960, then 7 yeare will yielde 6720: as these two fi∣gures doe shew.

〈 math 〉〈 math 〉

* 1.9But now sir, if I set forth 30 bushels of corne to another man for 7 yeare, agréeing

so that he shall sow euerye yeare the whole encrease of the corne, and I at the ende of those seuen yeares to haue the halfe of the whole increase: I would know how many bushels will there amount to my part sup∣posing the increase to be after y^e rate of the last question, for 30 bushels in one yeare, 360.

Mai.

In such a question you must haue so manye seuerall workinges, as there be yeres, as for example: In the first yere ••0 bushels yield 360: then to knowe the yiel∣ding of the second yeare, I must say: If 30 yield 360, how many yieldeth 360? Work by your rule, and you shal find 4320. Then say for the third yere: if 30 yield 360, how manye will 4320 yielde? you shall haue 51840, and so euery yeare multiplying the whole encrease by 360, and diuiding it by 30, the increase of the next yere wil amoūt, as these 7 figures (in y^e next page) do order∣ly declare: where I haue set 7 letters for y^e 7 yeres, of which the first is set without art because that is the increase whiche you doe presuppose: & the last number of eche other doth shewe the increase of the yeare that it standeth for; which the letters doe declare,

so that the increase of the seuen yeare, is 1074954240 bushels: how manye quar∣ters that is, and also how many wayes, you may by Reduction soone finde. 〈 math 〉〈 math 〉 Now with one question more I will proue you. If 6 Mowers doe mowe 45 acres in 5 dayes, how many mowers wil mow 300 acres in 6 dayes?

Sch.

If 45 acres doe require 6 mowers,

then 300 acres requireth 40. Now agayn: if 5 dayes require 40 mowers, then 6 dayes néedeth but 33 mowers.

Mai.

Why do you not make mention of the 2 that remaineth in the last diuision? for the last part of the question is wroughte by the Backer rule, where the first number 5, is multiplied into the seconde that is 40, whereof amounteth 200, whiche if you di∣uide by the thirde number 6, the quotiente will be 33, as you said, but then will there remaine 2, which cannot wel be diuided in∣to 6 parts: how be it, you may vnderstand by the sixt part of 2, the thirde parte of one mans work, which you must put to the 33, or else you may say, that 33 workemen wil ende all the 300 acres in 6 dayes, saue two mens worke for one day, or 2 dayes worke for one man. But such brokē numbers cal∣led Fractions, you shall hereafter more bet∣ter perceiue, whē I shal wholy instruct you of them.

Mai.

Yet one question more of field mat∣ters I will propone, and so I will make an end of this double Rule of 3.

Sc.

With all my hart sir I thank you, and I wil dispatche it as soone as I can, because

would faine sée the order of the nexte Rule of 5 numbers.

Maist.

Then this is my question, If 300 Pioners in 8 houres, will cast a trenche of 200 Rods: I demaund how manie Labou∣rers wil be able with a like trēche in thrée houres to entrench a Camp of 3500 Rods.

Sch.

I thinke I am nowe in the Backe-house diche, for I knowe not well whiche way to go about it: And besides that trulie I think I shall neuer come to prefermente that way my grouth is so small.

Ma.

You know not how God may raise you hereafter by seruice, into the fauour of your Prince, for y^e auaile of your Countrie. Example, Sir Francis Drake, as worthy a man as euer England bred, is not the tallest man, and yet hath made the greatest aduē∣ture for the honour of his Prince & Coun∣trie, that euer English man did.

Sc.

Sir, I thanke you for your good en∣couragement, my mind, though I be little, is as desirous of knowledge, as any other: I haue pondred now a little 〈 math 〉〈 math 〉 of it, & thus I set forthe the worke.

Saying if 200 Rod require 3400

300 men what shall 3400 Rod require: I multiplie 3400 by 300: and it yieldeth 1020000: which I diuide by 200 and my quotient is 5100 men.

Then must I say for my second work, if in 8 houres 5100 men be able to discharge it, how many shall performe the same in 3 houres? now if I should worke by y^e Gol∣den Rule of proportion forwarde, I should find a lesse nūber of men, because 3 houres is lesse then 8 houres: but because reasō tea∣cheth me that the lesser the time is, wherin y^e french must be made, the more Laborers I ought to haue, wherevpon I vse now the backer Rule as in example. And I haue in my quotiēt 5000. So many Pioners must I haue, to entrenche the camp in 3 houres.

Ma.

You haue answered the question ve∣ry artificiallie: And truelie I commende you for your diligence and apte vnderstan∣ding: and now according to my promise, I will (in whole numbers) giue you a little tast of the Rule of 3 compounded of 5 num∣bers.

The Rule of 3 compound of 5 numbers.

THis Rule of 3 com∣posed is distinct for most néedefull que∣stions into two se∣uerall parts or wor∣kinges: And there belongeth vnto it al∣wayes 5 numbers, wherof in this rule being the first part: the seconde number and the fift are alwayes of one nature and like denomination, which rule is to be wrought thus: you must mul∣tiplie the first number by the seconde: And that shalbe your diuisor: Then again, mul∣tiplie the other thrée numbers, the one by the other and their product shall be your di∣uidend.

And now according to my promise, wée will first work the question of weighte and cariage whiche I deliuered you in y^e double rule of 3: to be absolued by this Rule, which was this.

If the cariage of 100 lb. weight 30 miles

coste 12 d, what wil the carriage of 500 lb. weighte stande me in being carried 100 miles.

Then marke 〈 math 〉〈 math 〉 wel how these numbers stand: multiplie 100 by 30 as thys figure sheweth: And that number keepe for your diuisor:

Then multiplie the other 3. numbers the one by the other, and they amount as you see to 600000: which you shall diuide by 3000: your quotiēt is 200 d: Now you see it agréeth with the conclusion of the double Rule.

S.

Sir I thanke you most hartely: it is euen so.

Ma

Yet note this for a generalitie in thys Rule,* 1.10 loke what nature or denominatiō your middle number is: and of the like denomina∣tion or nature is alwayes your quotient.

Scholer.

Well nowe and it please you by your patience, I will sée howe I can ende, the question then nexte following of 30 Bu∣shels of Wheate sowed, yéelded in one yeare 360, how many then will 80 bushels yéelde in seauen yeare: and according to youre rea∣sons, I set my numbers thus:

which 201600 〈 math 〉〈 math 〉 I diuide by 30: and my quoti∣ente is 6720: bushels my de∣sire.

M

Yet one question more I will propound vnto you and so leaue this rule, till it please God hereafter, that I may make you worke it in broken numbers.

What comes the interest of 258 lb for fiue monethes after the rate: of 8 pounde taken in the 100 lb: for 12 monthes:

Sch

Sir as this is a question of gaines So will I warelie worke this question in hope one day to reape someting for my paines: and thus I propone it. But I beseech you if it bée not well set downe to shewe me myne er∣rour.

Ma.

Procéede you 〈 math 〉〈 math 〉 haue done verye wel

Sch.

Then I doubt not by the grace of God but to ende it: I multiplie 100 by 12 it yéel∣deth 1200: and the 3 other numbers multipli∣ed togither produceth 10 20: which I diuide by 1200: and my quotient is 8 poundes. Thē

according as you haue taught me heretofore, I turne the 720 lb. that is left: into shillings: and diuiding it by my firste number my quo∣tient is 12 s. So I answeare that the lone of 258 lb for 5 monthes, after the rate of 8 lb. in the 100 lb for a yere, comes to 8 lb 12 s.

Mai.

You say true, I commende your dili∣gence, now beholde the manner of the second part of this rule.

M.

In the second part of this rule of 3 com∣posed: the third number is like vnto the first. And the rule is to be wrought as thus: you shall now contrarie to the last rule multiplie the third number and the fourth togither: and that product shal be your deuisor: Then mul∣tiplie the fift by the seconde and the producte therof by the first: and that is y^e number that shall be diuided. For example I propond this question: for a proofe of my last question of in∣terest. A Merchant hath receiued 8 lb 12 s. for interest for 5 months terme, which he re∣ceiued after the rate of 8 lb. in the 100 lb. for a yere. The question is nowe how much mo∣nie was deliuered to raise this interest: Be∣hold therefore the 〈 math 〉〈 math 〉 maner howe the question is set foorth.

Scholer.

Sir I perceiue it verie wel: and according to the doctrine whiche you prescri∣bed for the working thereof: if it please you nowe it is set downe I thinke I can followe the worke.

M.

Nay staie a while, and afore you worke marke wel how I deliuer a reasō, for the per∣fect vnderstanding of this rule which is thus:* 1.11 if 8 lb. in 12 monethes do yéeld me 100 lb To take 8 lb — 12 s. for 5 moneths, muste néedes yéelde a great deale more.

So vpō the knowledge that I haue in this Arte, The first part of this rule is aunswera∣ble to the rule of 3 forwarde: And this latter part accordeth to the rule 3 backward.

S.

Sir I yéelde you most hartie thankes for these your last instructions, they haue giuen me great light into these two Rules, wherby I maye the better by deliberation conceiue how to vse them hereafter, whē occasion shall require.

M.

You say wel, go too now if you wil, and trie your cunning in the question:* 1.12 But thys note take with you by the way, in as much as here is mention made of shillings: turne all your mony as you worke, for your more ease in worke.

S.

If it please you to behold me a litle, I wil quicklie end it: for I haue but my first: my se∣conde: and my last number to be multiplied togither for my diuidende: And my third into my fourth for my diuisor: 〈 math 〉〈 math 〉

Whiche I diuide by 800. and my quotient is 5160 shillings, which in poundes yéeldeth 258 my desire.

M.

I will here for this time in whole num∣bers ende this rule, and wil instructe you in the rules of Felowship. You may at your cō∣uenient leasure, for your exercise worke the

same, by the rule of 3. at twice: And for your aide and encouragement therein, I set down here a profer how to apply it.

〈 math 〉〈 math 〉

The Rule of Felowship.

BVt now wil I shew you of the rule of fe∣lowship or Compa∣nie, which hath sun∣drie operations, ac∣cording to the diuers number of the com∣panie. This rule is sometime without difference of time, and sometimes there is in it difference of time. Firste I will speake of that without difference of time, of which let this be an example.

Foure Merchaunts of one companie made a banke of money diuerslie, for the first layde

in 50 lb, the seconde 50 lb, the third 60 lb, and the fourthe 100 lb, whiche stocke they occu∣pied so long, till it was encreased to 30••0 lb. Nowe I demaunde of you, what shoulde eache man receiue at the parting of this money.

Sch.

I perceiue that this rule is like the o∣ther, but yet there is a differēce, which I per∣ceiue not

Ma

Then wil I shew it to you. Firste by Addition you shall bring all the particular summes of the Merchaunts into one summe, whiche shall be the first summe in your wor∣king by the Golden rule, and the whole sum of the gaines by that stocke shal be the second summe. Now for the third sum, you shall set the portion of each mā one after an other, and then work by the Golden rule, and the fourth sum will shewe you each mans gaines: as in example.

The parcels of those foure Merchaunts make in one sum 240 lb: set that in the firste plare, the gaines in the second, 〈 math 〉〈 math 〉 and the firste mans portion of stocke in the place thus.

Now multiplie the second by the third, and it will be 90000, whiche you shall diuide by

240, and there wil ap∣peare 〈 math 〉〈 math 〉 375 lb. thus.

And that the gaines for the first man.

Now for the second man, set the 50 lb. that he brought, in the thirde place, and worke as before: and his part will 〈 math 〉〈 math 〉 be 625 lb. as this figure sheweth.

Likewaies for the thirde man set his mo∣ny 〈 math 〉〈 math 〉 which was 60 lb, and his parte of gaines wyll be 150 lb, as here appea∣reth.

And so for the fourth man, if you sette his summe whiche is 1000 lb, 〈 math 〉〈 math 〉 his gaines wil be 1250 lb, as the proofe wyll de∣clare.

Scholer.

This I perceiue: but is there a∣ny way to examine whether I haue wel done er no?

Maister.

That muste you doe by one com∣mon proofe which serueth to the Golden rule and al other insuing of the same: and that is this: Change the standings of the numbers, and set the thirde in the first place, the 4 in the seconde place, and the firste in the third place,

and they worke by the Golden rule, and if you haue done well, the fourth number now will be the same that was the seconde before. As for example, I will 〈 math 〉〈 math 〉 take the last worke whi∣che was this.

〈 math 〉〈 math 〉 Which to examine I alter as I saide, thus:

Nowe if I multiply the second number by the thirde, and diuide that that amounteth by the firste, then will the fourth number be 3000, 〈 math 〉〈 math 〉 whiche was the seconde before, as you sée here, whiche is a token, that I haue well done. But as in a single rule one proofe thus is sufficient, so in a rule where many operations be, you must turne euery of them as I haue done with this one.

Scholer.

Then for the 〈 math 〉〈 math 〉 proofe of the firste worke of this rule, I shold turn the numbers thus.

〈 math 〉〈 math 〉 And the seconde thus. And for the thirde thus. 〈 math 〉〈 math 〉 And in ech of them if the

working were true, the fourth number wyll be still 3000.

Ma.

Wel, nowe an other example will I put to you, not of gaines, but of losse: for one reason serueth for both.

If thrée Merchauntes in one shippe and of one fellowship, had bought marchandise, so that the first had laide out 200 lb, the seconde ••••00 lb, and the thirde 500 lb, and it chaunced by tempest that they did cast ouer board into the sea merchandise of the value of 100 poūd, howe muche shoulde eache man boare in this losse?

Scholer.

If I shal doe in this as you did in the other question, then muste I ioyne theyr thrée portions togither, 200, 300, and 500, whiche maketh ••000. Then saye I, if 1000 leefe 100, then shal 200 loose 20, and ••00 shal loose 30, & 500, shall loose 50, as by these thrée figures it doth appeare plaine.

〈 math 〉〈 math 〉

Ma.

Well sith now you haue done these I

wil propound a questiō of more importance, which shal make you not only y^e abler to vn∣derstād this Rule, but also it wil greatly aide you in the next rule of fellowship with time, if such néede be that your money be of diuerse denominations.

For this may not be forgottē in al such que∣stions, if the number be of diuerse kinds: you must by Reductiō bring it into one kind, that is to say to the leaste value that is named in y^e question. And likewaies shall you doe, if the time be of diuerse kinds, as some yeres, some monethes, wéekes and dayes, you shall make al months, wéeks or days according as y^e least name of time in y^e questiō is: As for example.

First in diuersitie of mony. Thrée cōpaniōs bought 2000 shéepe, and paide for them 241 lb. 13 s, 4. d. of which sum one paide 101 lb,* 1.13 10 s. The second paid 82 lb. 17 s, 10 d. And the third paide 57 lb, 5 s, 6 d: How many shéepe must each of them haue? Answere: The first shal haue 840. The second 686. And the third 474. And that must you worke thus.* 1.14

Firste considering that your money is of diuerse denominations, you shall (by Redu∣ction) bring it all into the smallest denomi∣nation

whiche is in it, that is to saye, pence, and so will the total sum bée 58000. pence.

Now, if you turne eache mans mony into pennies also, the firste mannes summe will be 24360 pence: The second mans summe 19894. pence. And the third mans mony wil be 13746. pence.

Now to know how many shéepe euery mā shal haue, let the whole sum of money that is 58000. pence, in y^e first place & in y^e secōd place set the number of shéepe, and then orderly in the thirde place set eache mans money, and then multiplying the thirde and the seconde summes togither, and diuiding that that a∣mounteth by the firste, there will appeare the number of shéepe that eche man ought to haue: as these thrée figures do shewe, 〈 math 〉〈 math 〉

Scholer.

Why doe you set the mony in the first place, séeing in the question you saye,

2000 sheepe cost 58000 d? & not thus, 58000 d cost 2000 sheepe.

Mayster.

You remember, I taught you at the beginning of this Goulden rule, that the firste and thirde numbers must bée of one name, and of like thyngs: and euermore the number that the question is asked of, must bée sette in the third place. Now is the question playnely this: If foure men bought 2000 sheepe for 58000 pence, howe many sheepe shall each man haue?

But seing in this question there ought more respect to be had to the summe of mony, than to the summe of the persons, (for in y^e sūmes of mony is there proportion toward y^e sheepe, and not in the number of persons) therefore must wée turne the question thus.

If 18000 pence bought 2000 shéepe, how many did 24360 s buy? Agayne, how many did 19894 d buy? & how many bought 13746 pens.

Scholler.

I perciue it reasonable, and so shall I doe in all like questions.

Mayster.

Euen so, But for easinesse of the work marke this:* 1.15 When soeuer the first and second numbers haue ciphers in the first places, you may bothe in the multiplication

and in the diuision leaue out those ciphers, so that you leaue out like manye out of bothe summes, as in this question the first number 58000 hath thrée ciphers, and so hath the se∣conde that is 2000: therefore caste awaye their ciphers, and so will the first number bée 58, and the second 2: set them in their places, and worke according to the rule, and you shal perceiue that it wil be al one, sauing that this is the shorter and easier way, as these thrée fi∣gures do shew.

〈 math 〉〈 math 〉

And this you sée is both easier, and also the more certaine waye to know the answere to this question.

S.

Truth it is as you say: but sir, me séemeth I might aske a further question here, not on∣lie how manye shéepe eche man should haue, but also what euerie shéepe cost.

Maister.

That question doeth not onelye

belong to this rule, but may also be discussed by Diuision, especiallie if the questions num∣ber be one onelie: as thus. Diuide the totall summe 58000 pence, by 2000 (or 58 by 2, omitting the ciphers) and the quotient wil be 29 pence, that is 2•• s, 5 d, howbeit, by this rule you maye doe it, and beste when the number of the question doth excéede 1: as if I shoulde aske this question, 2000 shéep•• coste 58000 d,* 1.16 howe 〈 math 〉〈 math 〉 muche did 20 coste? Then shal I set my figure thus.

And doing after the rule, there wil amoūte 580 pence, that is 2 lb 8 s 4 d: the price of one score: But if you wil vse that easie waye that I did teach you, you may 〈 math 〉〈 math 〉 change the firste and seconde number thus.

Thus doe you perceiue the vse of the rule without time.* 1.17 And that you may as wel•• perceiue the same with diuersitie of time, I propose this example.

Foure Merchants made a common stocke,* 1.18 whiche at the yeares ende was encreased to 35145 lb. Nowe to knowe what shall be ech mans portion of gaines, you muste know

eache mans stock and time of continuance.

The first man of these foure laide in 669. lb which hée did take from the stock agayne, at the end of 10. moneths. The second man layd in 810. lb. for 8. moneths. The thirde layde in 900. lb. for 7. moneths. And the fourth layd in 1040. lb. for 12. moneths.

This question shall you examin as you did the other before, sauing that where as in the third place of the figure you did set eche mans summe alone,* 1.19 here you shall set the same bée∣ing multiplyed by the number of their time & likewise in the firste place of the figure, you shal set that number which amoūteth of their whole summes so multiplyed by their time, & added into one whole summe as thus.

The first mans summe is 669 lb. which I multiplye by 10. (that was the number of his time) and it maketh 6690. The second mans summe 810. lb. multiplyed by 8, (which was his time) make 6480. The third mans sum 900. lb. multiplied by 7 (for that was his time yeldeth 6300. The fourth mans summe was 1040 lb and his time 12. multiply the one by the other, and and it will be 12480.

These foure summes thus multiplyed by their time, must be set orderly in y^e third place

of the figure: and in the first place must bée set the whole summe of all foure, whiche is 31950, and the gaine must be in the second place, which is ••5145. Now to end the que∣stion, I say firste: If 31950 did get 35145, what did 6690 get? 〈 math 〉〈 math 〉 Answere, 7359 lb, as by this figure ap∣peareth.

Likewise the second man had to his part 7128 lb, the third must haue 6930 lb And the fourth mā shal haue for his part 13728 lb as these figures doe partly declare.

〈 math 〉〈 math 〉

Scholer.

This I like verye well: but what proofe is there of this worke?

Mai.

The same that I taughte you for the other. Howbeit,* 1.20 there is vsed both for this worke and the other also this manner of proofe, to adde all the portions togither,

and if they agrée to the whole summe, then seemeth it well done: but this is no sure rule.

Sch.

Yet will I prooue it in this example. The foure parcels are these, 〈 math 〉〈 math 〉 which if I adde togither, there will amount 35145, and that was the whole summe: so is this rule true here.

Maister.

And so will it be still when the worke is truely done.

* 1.21But if you lift to sée it prooued false, take 10000 lb from the fourth man, and put it to any of the other 3, and then be yée sure that you haue not done well, and yet wil the proofe allow it, for the Addition will still be all one.

Sc.

It must néedes be so: but what haue I now to learne?

Maist.

There are many other excellent partes behinde, of which I will not, as nowe, make mention because that without the knowledge of Fractions, they cannot be duely taught, and much lesse vnderstanded. Therfore wil I propose to you two or thrée questions more, whereby you may practise the better the feate of the rule of felowship,

(that thereby you may better perceiue y^e vse of all other) & so make an end for this time.

There is in a Cathedral Church ••0 Can∣nons, and 30 Vicars,* 1.22 those maye spende by yere 2600 lb, but euery Cānnon must haue to his part 5 times so much as euery Vicar hath: howe muche is euery mans portion say you?

Sc.

I pray you make the aunswere your selfe, so shall I perceiue best the meanes to aunswere to such other like.

Mai.

In this question you must doe as in those that haue diuersitie of time, for here is diuersitie of portions: Therefore shall you multiplye the number of the persons by their difference of portion: (as you did in the other by time) Then must you mul∣tiplie the ••0. (which is the number of Can∣nons) by 5, (for that is the number of their portion) so will it be 102: Then ••0, (that is the number of Vicars) by 1, (that is the number of their portion) and it will be 30: put those two summes togither, and they make 1••0: then say thus: If 130 spēd 2600 lb, what maye 100 spende? The rule sheweth 2000 lb.

Againe for Vicars: If 130 spende 2600

pound, what may 30 spend? Answer 600 lb as these figures shew, 〈 math 〉〈 math 〉

But if euery Cānon should haue so often times 4 lb as the Vicar should haue 3 lb thē should I multiplie 20 by 4, (that were 80) and 30 by 3 (that were 90) and then both were 170. Then should the figures be sette thus.

〈 math 〉〈 math 〉

But this sort is to hard for you, by reasō of the Fractions, therefore I will let it rest to that place. And by this rule you sée what the 20 Cannons may spend, which summe if you diuide by 20, you shall sée eche Can∣nons portion: and so of the Vicars, if you diuide their summe by 30, the quotient wil declare euery Vicars portion.

Notes

• * 1.1

  The third number.

  The seconde number.

  The first number.
• * 1.2

  The bac∣ker rule.
• * 1.3

  Question of buying cloth.
• * 1.4

  Question of prouisi∣on tou∣ching a••∣a••••ate.
• * 1.5

  Note.
• * 1.6

  The dou∣ble Rule.
• * 1.7

  Of cariage.
• * 1.8

  Question of sovving.
• * 1.9

  Question of Corn.
• * 1.10

  Note.
• * 1.11

  Note.
• * 1.12

  Note.
• * 1.13

  Question of Sheepe.
• * 1.14

  Solution.
• * 1.15

  Note.
• * 1.16

  ☜
• * 1.17

  The rule of felovv••h••p vvith time.
• * 1.18

  Question of a banke.
• * 1.19

  ☞
• * 1.20

  Another proofe.
• * 1.21

  Note the imperfecti∣on of this kinde of proofe.
• * 1.22

  Question of Canons.