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.

A Rule.

This beynge deen, you shall mul∣tiplie

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

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

Lib: Clothes. Lib.

〈 math 〉〈 math 〉 And thā as you see, multiplie y^• thirde nomber, which is. 300. l••. by. 2. whiche is the second nōber, and therof cōmeth 600. the which. 600. you must diuide by the first nōber. 15. and you shal find in your quociēt 40. whiche is. 40. clo∣thes, and so many clothes shall I buye for. 300 .l. as appereth by practise here aboue written. And here you muste marke that the first nōber & the thirde in this questiō be of one denominaciō, and like wise the second & the fowerth which you haue found are of one sem∣blaunce: but in case that the first nom∣ber and the third in any question: bee not of like denominacion, you muste in workyng bryng them into one, as in this exāple folowing. If. 12. nobles do gaine me 6. nobles, how many no∣bles wil. 48. poundes gatne me: Here you se that y^• denominaciō of the first

nomber is nobles, and the denomina∣cion of the thirde, is poundes, where∣fore, before you doe procede to worke by the rule of three, you muste firste turne the poūdes into nobles in mul∣tipliyng. 48. poundes by three nobles and they make. 144. nobles, for that there is in euery pounde of money. 3. nobles, or otherwise if you will, you maie bryng the first nomber beyng. 12 nobles, into poundes, by diuiding thē by. 3. and thus shal your first and third nombers, bee brought into one deno∣minacion. Then shal you sette doune your. 3. nombers in order thus.

If. 12. nobles doe gaine me. 6. no∣bles, what shall. 144. nobles gaine? the which. 144. are the nobles which are in. 48. 〈◊〉〈◊〉. Then multiplie the third nomber. 144. by the seconde nomber 6. and thereof cometh 864. the whiche you muste diuide by. 12. nobles, and thereof commeth. 72. nobles.

But here it maie perchaunce make some menne masse, to see all the three

nombers in this rule of three, to bee of one denomination, whiche can not o∣therwise bee dooen, if you reduce the third nomber, to the denominacion of the firste. But if you will reduce the firste nomber, to the denomination of the thirde, that is to saie the. 12. nobles into poundes, then shall the firste and the third nombers onely agree in one denominacion, and the fowerth nom∣ber which you seke, shalbe of the same denominacion as is the seconde, as in the former example. If. 12. nobles doe yelde me. 6. nobles, what will. 48. poundes yeld me: first you shall diuide 12. nobles by three to bryng theim in poundes, and thei shall bee like to the thirde nomber, whiche is also poun∣des, then will thei stande thus. 〈 math 〉〈 math 〉

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

then worke by the rule as before. 〈 math 〉〈 math 〉

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

the backer rule of three.

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

committe an euidente and open er∣rour.