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THE rule of three is the chie∣fest, the moste profitable, and the moste excellente rule of all the rules of Arithmetike. For al other rules haue nede of it, and it passeth all the other, for the whiche cause it is saied, that the Philosophers did name it the golden rule. And after others opinion and iudgemente, it is
called the rule of proportions of nom∣bers. But now in these daies, by vs it is called the rule of thrée, be cause it re∣quireth three nombers in his operaci∣on. Of the whiche three nombers, the twoo first are set in a certain propor∣tion. And in suche proportion as thei be stablished, this rule serueth to finde out vnto the third nomber, the fourth nomber to hym proportioned, in suche sort as the seeonde is proportioned vn∣to the firste. Not for that, that the fo∣wer noumbers, nor yet the three, are or bee proportionall, or set in one pro∣portion, but suche proportion, as is from the firste to the seconde, ought to bee from the thirde vnto the fowerth, that is to sate, if the seconde noumber dooe contain the firste, twoo tymes or more, so many tymes shall the fourth nomber containe the thirde. And note well that the firste noumber, and the thirde in euery rule of three, oughte and must bee alwaies semblable, and of one condition. And the second nom∣ber,
and the fowerth muste likewise bee of one semblaunce and nature.
And are dissemblaunte, and contrarie to the other twoo noumbers: that is to saie to the firste, and the thirde. And if you dooe multiplie the firste by the fo∣werth. And the seconde noumber by the thirde. The twoo multiplications will bee egall. Likewise if you diuide the one sembleaunte by the other, that is to saie, the thirde noumber by the firste. And likewise the one dissemble∣aunt by the other: that is to saie, the fowerth nomber by the second (which are dissembleaunt to the other twoo nombers) your twoo quocientes will be egall
The stile of this rule in thus, you * 1.1 muste sette doune your three noum∣bers in a certaine order, as by exam∣ple here vnder shal appeare. And then multiplie the thirde noumber, by the seconde. And the producte thereof you must diuide by the firste noumber, or otherwise, diuide the firste noumber
by the seconde. And the quocient ther∣of shalbee diuisor also vnto the third nōber that is to saie, the thirde nomber shall bee diuided by the quotient of the foresaied diuision, that is of the firste noumber diuided by the seconde. Or otherwise diuide the second noumber by the firste. And that whiche cōmeth into your quotiente, you shall multi∣plie it by the thirde nomber. And thus shall you haue the fowerth noumber, whiche you seke for.
IF, 8. be worthe. 12. what are 14. worth after the rate, or els if. 8. require. 12. for his pro¦portionall, what will. 14. de∣maunde? The whiche three noum∣bers maie conuenientlie bee sette in soche order, as hereafter doeth appere.
If. 8.-12.-14. multiplie the thirde nomber. 14. by the seconde, whiche is. 12. And thereof commeth
(for the whole producte of this multi∣plication. 168. the whiche (as the rule teacheth) you muste diuide by the first nomber, that is to saie by. 8. and ther∣of commeth. 21. And so muche are the 14. worthe. This is the waie, whiche is moste vsed.
Otherwise diuide. 8. by. 12. whiche you can not doe, for thei are 8/72. where∣fore abreuie. 8/12. and thei are 2/9. for your quotient, then diuide the thirde nom∣ber. 14. by the saied ⅖, and you shall haue. 21. as before. Orels diuide the seconde nomber. 12. by the firste nom∣ber. 8. thereof commeth. 1 ½. the whiche 1 ½ you shall multiplie by 14. and ther∣of will come. 21. as is aboue saied, and thus muste you dooe of all other. And although, that the nōbers of this rule maie bee founde in three differences, for sometymes thei are whole nom∣bers and broken together, sometymes broken and broken together, and som∣tymes all whole nombers, if thei bee whole nombers, you muste doe none
otherwise, then you did in the last erā∣ple. But in case thei be broken noum∣bers, or broken and whole noumbers together, the maner and waie to dooe theim, recetueth a certaine variacion, and difficultee, according to the vari∣etie of the noumbers, the whiche ope∣ration easeiy to do, and vnuariable, this rule teacheth.
The three noumbers beyng sotte doune, according vnto the order of the whole noumbers aforesaied, without any broken nomber, let. 1. bee put al∣waies vnderneath euery whole nom∣ber, with a line betwene them fractiō wise, as thus 8/1. and that. 1. is denomi∣natour to euery soche whole nomber. When you haue whole nomber and broken, thei must bee reduced and ad∣ded with their broken nomber, and if there bée broken nomber without any whole nomber, thesame broken must remain in their estate.
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.
If. 15. shillynges worthe of Wine wil serue for the ordinary of. 46 men when the Tonne of Wine is worthe 12. poundes: for howe many menne will the same. 15. shillynges suffice, when the Tonne of Wine is worthe but. 8. poundes? It is certain, that the lower the price is, that the Tonne of Wine doeth coste, and so many more persones will the saied. 15. shillynges in Wine suffice. Therefore set doune your nombers thus, if. 12. poundes suffice. 46. menne, how many will. 8. poundes suffice, you muste multiplie 46. by. 12. and thereof commeth. 552. the whiche. 552. you shall diuide by. 8. and thereof commeth. 69. and vnto 69. menne will the saied. 15. shillyn∣ges worthe in Wine suffice, when the Tonne of Wine is worthe but eight poundes, as hereafter dooeth appeare by practise.
〈 math 〉〈 math 〉
Likewise, a messenger maketh a iourney in. 24. daies, when the daie is but, 12. houres long: how many daies shal he be vpon the same iourney, whē the daie is. 16 houres in length? Here you must perceiue, that the more hou∣res are in a daie, the fewer daies will the messenger bee in goyng his iour∣uey. Therefore write doune your nō∣bers thus, as here you maie se. 〈 math 〉〈 math 〉
And then multiplie 24. daies by. 12
houres, and thereof commeth. 288. di∣uide the same. 288. by the thirde nom∣ber. 16. and you shall finde 18. the whi∣che is. 18. daies, and in so many daies will the messenger make his iourney when the daie is. 16. houres long.
Likewise, when the Bushell of wheate doeth coste. 3. shillynges, the penie loafe of bread waieth. 4. lib.
I demaunde what the same penie loafe shall weye, when the bushell of wheate is worthe but twoo shillyn∣ges: here is to bee considered, that the better cheape the Wheate is, the hea∣uier shall the pennie loafe wiie, and therefore write doune your. 3. nom∣bers, thus. 〈 math 〉〈 math 〉
Then multiplie. 4. lib. whiche is the secōde nomber, by the first nomber 3. and thei make. 12. the which. 12 you
shall diuide by the thirde nōber. 2. and thereof commeth. 6. lib. and so muche muste the penie loafe of breade waye, when the bushell of wheate is worthe but twoo shillynges, as maie appere.
And nowe, accordyng to my for∣mer promise, shall followe the seconde parte of Arithmeticke, whiche teacheth the workyng by Fractions.
Regul.