The vvell spryng of sciences whiche teacheth the perfecte woorke and practise of arithmeticke, bothe in whole nombers and fractions, with suche easie and compendious instruction into the said arte, as hath not heretofore been by any sette out nor laboured. Beautified with moste necessary rules and questions, not onely profitable for marchauntes, but also for all artificers, as in the table doeth partlie appere: set forthe by Humfrey Baker citezeine of Lo[n]don.

¶ The 13 Chapter treateth of the Rule of Alli∣gation.

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

Example.

1 A Goldsmythe hath three sortes of Golde. The fyrste is worth thir∣tye Crownes the pounde weyghte: The secende is worthe 36 Crownes. And the thyrde is worthe 45. crow∣nes, and of these three sortes he will make a Scepter of sixe pound weight, whiche shalbe worthe 40, Crow∣nes the pounde. I demaunde howe muche he muste take of euery sorte?

Answere: first you must set downe the nōbers whereof you wyl make the Al¦ligation (which are 30. 36. 38, and 45) orderly the one vnder the other, as if you shoulde make of them an addiciō: and the common nomber whereunto you will reduce them, shall you set on the left hand, which common nomber in this example is 40. Then marke what summes bee lesser, then that common nomber, and whiche bee greater, and wyth a draught of your penne, euermore lynke two nombers togither, so that the one be lesser then that common nomber, and the other greater then he. For 2 greater nor twoo smaller nombers maye not bee linked together, for they wyll eyther be lesser, or els greater then the com∣mon nomber: but one greater nom∣ber, and one smaller maye be so mi∣xed, that they will make the common nomber. And two greater or twoo smaller nombers, can neuer make the common nomber in dewe order,

as here after shall appeare.

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

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

40 is 10, whiche I set against 45, that he is lynked wythall, and the diffe∣rence of 45, aboue 40 is 5. whiche I haue set against 30.

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

Multiplie and diuide, and you shall fynde one pounde 1/7: so muche muste I haue of the fyrste price.

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

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

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

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