The welspring of sciences, which teacheth the perfecte worke and practise of arithmeticke both in vvhole numbers & fractions, with such easie and compendious instruction into the saide art, as hath not heretofore been by any set out nor laboured, : Beautified vvith most necessary rules and questions, not onely profitable for marchauntes, but also for all artificers, as in the table doth plainely appere..

¶A Rule.

This beyng doen, you shall mul∣tiplie

the denominatour of the firste nomber, by the numeratour of the se∣conde, and the producte thereof againe by the numerator of the third nomber 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 cometh of this multiplica∣cion 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 the same price, that the twoo clothes did cost, sette doune your three nombers thus.

Lib. Clothes. Lib.

15.	2.	300.	1
		2	600
		600	155	(40
			1

And thā as you sée, multiplie the third nomber, whiche is .300. li. by .2. which is the second nōber, and therof cometh 600. the whiche .600. you must diuide by the first nōber .15. and you shal find in your quotiēt 40. whiche is .40. clo∣thes, and so many clothes shall I buye for .300. li. 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 likewise the seconde & 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 will .48. poundes gaine me: Here you se that the denominaciō of the first

nomber is nobles, and the denomina∣cion of the thirde, is poundes, where∣fore, before you dooe procede to worke by the rule of three, you muste firste turne the poūdes into nobles in mul∣tiplying .48. poundes by three nobles and thei 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 shall you sette doune your .3. nombers in order thus.

If .12. nobles doe gaine me .6. no∣bles, what shall .144. nobles gaine? the whiche. 144. are the nobles which are in .48. li. Then multiple 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 muse, to see all the three

nombers in this rule of three, to bee of one denominacion, 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 denominacion of the thirde, that is to saie the .12. nobles into poundes, then shall the firste and the thirde 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.

Poūdes.	Nobles.	Poūdes.
4.	6.	48.
		6.		Nobles.
		288.	288	(72
			44

There is yet a more exacte waie to woorke in this rule of three, whiche is thus. You must marke if the third and first nombers in the rule of thrée, maie bee bothe diuided by one like diuisor: the whiche 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∣nes doe buye me .44. yardes of clothe, howe many yardes shall I haue for 120: here you maie 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.

Crounes.	Yardes.	Crounes.
5.	44.	12.
	12.
	88.	3
	44	528 (105 ⅕
	528	355

Multiplie .44. by .12. and thereof commeth .528. diuide thesame .528. by 5. and you shall finde in your quocient 105. ⅗. 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 woorkyng by the rule of three, but for that thei require the knowledge of fraccions, and because thei are not so easie as this first waie, whiche is common, therefore content your selues with this same, vntill you haue tasted the fraccions, the whiche by gods helpe I intende to set forth in seconde part of this boke, incontinen∣tly 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, whereof wee haue nowe treated. For in the directe rule of thrée the greater the thirde noumber is, so moche the greater will the fowerth be But here in this backer rule it is con∣trariwise, for the greater the third nō∣ber is, so moche lesser will the fowerth bee. Then, where as in the rule of .3. directe, the thirde 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 thesame by the thirde, and the nomber whiche commeth in the quociente, an∣swereth to the questiō. For soche prac∣tise commeth often tymes in vse: In soche sorte, that if you woorke the∣same by the rule of three directe (not hauyng a regarde vnto the Proposi∣cion of the question) you should then

commit an euident and open errour.