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THe Rule of falsehod is so named not for that it teacheth any de∣ceyte of falsehode, but that by fayned nombers taken at all aduen∣tures, it teacheth to finde out the true nōber that is demaunded. And this (of all the vulgare Rules whiche are in practise) is the most excellent: this rule hath two parts, the one is of one false position alone: the other is of two po∣sitions as here after shall appeare.
Those questions whiche are done by false positions, haue theyr opera∣tions, in a maner like vnto that of the rule of three, but onely that in the rule of three, we haue three nombers knowen, and here in this rule wee haue but one (I meane that commeth in operation) vnto the likenes where∣of we muste diuise two other, the one multiplying, and the other diuidyng, as by example.
1. I haue deliuered to a banker a certein summe of pounds in money, to haue of him by the yeare 6 li. vpon the 100 li. And at the ende of 10 yeres he payed me 500 pounde for all bothe principall an gaine. I demaund howe muche was the principall summe that I deliuered at the fyrste. Here you see that there are diuers termes: but the chiefe to worke with all is 500 li. whiche commeth of the other nom∣bers, that is to saye, of 10, and 100 for of them is compounde the tenour
of the question, the practise whereof is thus.
Let vs faine a nomber at pleasure, and with the same let vs make oure discourse, euen as though it were the principalll summe that we seke for. As by Example. Suppose that I de∣liuered hym at the firste 200 pounde, the whiche were worthe to me in 10 yeares 120 pounde after the rate of 6 vpon the hundred pound. Then 120 pounde added with 200 pounde. Do make but 320 li. and I must haue 500 li. Thus you see that I haue three termes for the rule of three: the one which shall conteine the Question the other two, which I haue 200, & 320: in such sorte, that 320, ought ta haue su∣the proporcion to 200, as 500 hat vnto the nomber that I seeke: that is to say, vnto the true principal summe, then must I haue recourse vnto the rule of three, after this sorte, saying.
If 320. pound become of 200. pounde of howe muche shall come 500. pound Multiply 500: by 200. and thei are 100000. the whiche you shall diuide by 320. pounde and thereof commeth 312. pounde. ½ whiche is the summe that I deliuered at the firste, and thus, this rule hath some congruence with the double rule of three.
2. I haue a Cesterne with 3. vnegal cockes conteining 60. pipes of water: And if the greatest cocke bée opened, the water will auoyde cleane in one houre, at the seconde yt will anoyde in twoo houres, and at the thirde it will require three houres. Nowe I demande in what space wil it auoide, all the cocks beyng set open. Suppose that it wyll auoide in halfe an houre, that is to saie, in 30. minutes. Then muste there auoide at the firste cocke, the ½, whiche is thirtie pipes, and by the seconde cocke the ¼ whiche is 15, pipes, and by the thirde cocke the ⅙, that is tenne pypes, all the whiche
summes beeyng added together doeo make fiftie fiue pipes, but it shoulde be sixtie pipes. Therefore I saie by the rule of thrée, if fiftie fiue pipes doe voide in thirtie minutes: in how ma∣nye minutes will sixtie pipes voide? Multiplie and diuide, and you shall finde thirtie twoo minutes 40/55. And in that space will the water auoide if all the cockes be set open.
THe summe of this rule of two false positions is thus, when any question is proponed ap∣pertaining to this rule. First imagine any nūber at your pleasure, which you shal name the first position, and with the same shall you worke in stede of the true number, as the questi¦on doth import, and if you sée that you haue missed. Then is the last number of ye worke either to great or to litle, yt shal you note for to be the first errour,
in the which you haue missed with the signe of more, or lesse, whiche signes shalbe noted with these figures, 4••:—This figure 4: betokeneth more, and this plaine line—signifieth lesse, that is to saie the one signifieth to muche & the other to little: then begin again, & take an other number, whiche shal be the second position, and worke by the question as before, if you haue missed againe, note the excesse or want, for that is the seconde errour. Then shall you multiplie the first position by the second errour crossewise, and againe the seconde position by the first errour (and this must alwaies be obserued) & keepe the two productes: then if the si∣gnes be both like, that is to saie, either both to muche, or both to little, abate the lesser producte frō the greater and likewise, you shall substract the lesser error from ye greater, & by the remaine of those errours, you shal diuide the re∣sidue of the products: the quotient shal be the true number that you séeke for.
But if the twoo signes be vnlyke, that is to say, the one to muche and the o∣ther to lyttle, then shall you adde those productes together so shall you also adde bothe the errours together, and by the summe of those errours, diuide the totall summe of both the products: the quotient shall be likewise the true number that the question seketh, and this is the wholerule, as by example. ••3. A manlying at the point of death, saied that he had in a certain Coffer a hundred duckets, the which he bequea ched to 3. of his frends by him named, after this sorte. The first must haue a certain portion, the 2. must haue twise so manie as the first abatyng 8. Duc∣kets & the third muste haue 3. times so many as the first, lesse by 15 Duckets. Nowe, I demaunde howe many eue∣rie of them must haue. Answere: First I doe imagine that the firste man had thirtie Duckets, then by the order of the question, the seconde shoulde haue fiftie twoo: and the third seuentie fiue.
These three summes beyng added to∣gether doe make 157: and I shoulde haue but a hundred soe that this fiirste errour is to much by fiftie seuen, then I note a parte the first position thirtie, with hys errour fiftie seuen, to muche after this sorte thirtie. 457.
Therefore I prosecute my worke and I suppose that the firste hadde twentie foure, then by the orderder of the que∣stion, the seconde should haue 40. and the third, fiftie seuen, these three sum∣mes beyng added together, doe ma•••• 121. and I must haue but a hundred, so the seconde errour, is to muche by 21. Therefore I note 24. 421. vnder the thirtie. 457. as you maye see in the margent of the nexte side following. Then I multiplie crossewaies, thirty (which is the firste position) by twelue which is the seconde errour, and ther∣of commeth 630. likewise I multiplie twentie and four (which is the second position) by fiftie seuen, whiche is the firste errour, and I finde 1368: then
because the signes of the errours are 〈 math 〉〈 math 〉 bothe like: that is to saye muche, I must therfore sub¦stracte 630. from 1368. and ther wil remain 731 which is the diuidende: again I must sub∣stract the lesser er∣ror from the grea∣ter, that is to wit, 21. out of 57. and there will remain 36. whiche shalbee my diuisor. Thys done I diuide 738 by 36 and the quotient will bee 20. 〈◊〉〈◊〉. The which 20. ½ is the iust number of the duckets that the first man had for his parte, so consequently the seconde manne had thirtie three Duckets, and the thirde fourty six ½, as by proofe in the margent maie appeare.
The like number wil also appeare. in case the errours wer both to little, as in makyng the twoo positions by 18. and 20. where you shal finde the 〈 math 〉〈 math 〉 two errours both to little, the firste will bée to lyttle by 15. and the se∣conde to little by 3. as by perusing this worke in the margent you shal well perceiue.
Againe if one of the errours were to muche, and the other to little, yet shall I haue the true number, as be∣fore: As if the twoo positions were 24. and 20. I shall finde that the firste errour will be 21. to muche, and the seconde wil be three to little: Therfore I multiplie twentie foure by thirtie, crossewaies, thereof commeth 72.
Likewise I multi∣plie twentie by 21. 〈 math 〉〈 math 〉 the product will be 420: These twoo summes 72. and 420, I adde both together because ye signes of the er∣rours bee vnlyke, and thei mak. 492. the whiche shalbee my diuidende, and againe, I addde the lesser errour three, with the greater errour 21. and they mak 24. for my diuisor, then diuiding foure hundred ninetie two by twenty foure, the quotiet wilbe twentie ½: as in the margent doth plainlie appeare.
And now because you shall not for∣get this parte of the rule, learne thys briefe remembraunce followyng.
The signes both like substraction doe require And vnlike signes addition will desireThe meanyng whereof is thus yf both the errours haue lyke sygnes, then muste the diuidende and the di∣uisor bee made by substraction, as is taught before, and if those sygnes bee vnlike, then muste I by addition ga∣ther the diuidende, and the diuisor, as I haue done in this laste example.
4. A man hath twoo siluer cups of vnegall waight, hauynge to them bothe, but one couer, the waight whereof is fiue ounces, if the couer be putte to the lesser cuppe, it wyll bee in double proportion vnto the waight of the greater, and the couer béeyng put to the greater cuppe, will bee in tri∣ple proportion, vnto the waight of the lesser. I demaunde what was the waight of euery cuppe.
Suppose that the lesser cuppe dyd wayghe seauen ounces, then wyth the couer it muste waigh twelue, and this waight shoulde be in double pro∣portion vnto the greater, therefore
the greater muste waigh six ounces 〈 math 〉〈 math 〉 adde vnto it 5 oun¦ces for ye couer, all wil be 11. ounces, but it should be 21 for to haue it in triple proportion, vnto 7. which re∣presēteth ye weght of the lesser cuppe So that this first errour is to little by 10. which you shall note after seuen in this sorte. 7.-10.
After you shall suppose some other number, as 9. and make the like work as before, so shal you finde 15. to little, for the second errour, which you shall put behind nine, and then worke with the reste as aboue is sayed, and you shall finde that the lesser cuppe wayed three ounces, and consequentlye the greater foure ounces.
5. One man demaunded of ano∣ther
in a mornynge what a clocke it was, the other made hym thys aun∣swere, if you doe 〈◊〉〈◊〉 (saieth hee) the ½ of the houres whiche he passte synce mindnight, with the ⅓ of the houres whiche are to come vntill noone, you shall haue the iuste houre, that is to saie, you shall knowe what a clocke it was: Suppose that it was 4•• clocke in the mornynge, so shoulde there re∣maine 8. vntill none: then I take the ••/4 of 4. whiche is 1. and the ⅔ of 8. whiche is 5. ⅓, and I adde them together, so I finde 6. ⅓ and I supposed but 4. ther∣fore thys first errour is to muche by 2. 〈◊〉〈◊〉, which I note after my position thus 4. † 2. ••/3: then againe I suppose an o∣ther number, that is to saye nine, soe shoulde remaine but 3. houres vntill noone., I take the ¼ of 9. and the ⅔ of 3. whiche is 2. ¼ and 2. these I adde to∣gether and they make 4. ¼: but I sup∣posed that it was 9. therfore the second errour is 4. ¾ to little whiche I note behinde my position thus. 9. † 4. 2/4.
And then I multiplie crossewyse, as before 〈 math 〉〈 math 〉 is taught, and bicause th•• sygnes of the er∣rours are vnlike, that is to saye, the one to muche, and the other to lyttle, therefore in thys woorke I muste adde the products, and they will bee fourtie. Likewise I adde the errours, and thei be seuen, 〈◊〉〈◊〉. Then I diuide fourtie by seuen 1/12, and thereof commeth fiue houres 11/17, and that houre it was in the mornyng.