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 thirde Chapter treateth of ab∣breuiation of one greate broken number into a lesser broken.

ABbreuiation is asmuch as to set downe, or to write a brokē nūber by figures of lesse sig∣nification, & not diminishing y^• value thereof. The whiche to doe, there is a rule whose operation is thus, diuide the numerator and likewise the deno∣minator, by one whole number, the greatest y^t you maye fynde in the same broken number, and of the quotient of that numerator, make it the nume∣rator, and likewise of that of the de∣nominator, make it your denomina∣tor, as by example.

1. If you wyl abbreuiat 54/8••, you shal vnderstande that the greatest whole number that you maye take, by the which you maye diuide the numera∣tor & denominator is 27, which is the half of y^• numerator, & that is a whole number, for you can not take a whole

number oute of the denominator. 81. but that there will bee either more or lesse than a whole number, therefore if you diuide 54. by 27. you shall 〈 math 〉〈 math 〉 finde 2. for the nu∣merator, likewise if you diuide 81. by 27. you shall finde 3. for the denomi∣nator. then put. 2. ouer the 3. with a line betwene thē, and you shall find ⅔ and thus by this rule the 53/81 are a∣breuted vnto ⅔, as appeareth in the margent, and so is to be vnderstande all other.

¶ The forme & maner how to finde oute the greater number, by the which you mai wholy diuide, y^• nu∣merator & denominator (to thende y^t you may abreuiat them) is thus.

First, diuide the denominator by hys numerator, and if anye number doe remaine, let your diuisor be diui∣ded by the same number, and so you must continue vntill you haue so di∣uided y^• there maye nothing remaine, then is it to be vnderstande, that your last diuisor (wherat you did ende, and that o did remaine after your last di∣uision) is the greatest number, by the whiche you must abreuiat, as you did in the laste example, but in case that your last diuisor be 1. it is a token that the same nūber can not bee abreuied. Example, of 54/81 diuide 81. (whiche is the denomination) by 54. which is his numerator, and there resteth 27. then diuide 54. by 27. and there remaineth nothinge, wherefore your last diuisor 27. is the number, by the whiche you must abreuiat 54/81 as in the laste exam∣ple is specifyed.

¶ An other stile of abbreuiation.

2. Mediate the numerator, and al∣so the denominator of your fraction in case the noumbers be euen, that is to saye, take alwayes the halfe of the numerator, and likewise of the deno∣minator, and of the mediatiō or halfe of the numerator, make the nu∣merator, also of ½ the denominator, make your denominator, and so con∣tinue as often as you may in takinge alwayes the ½ of the numerator, and semblablie of the denominator, or els see if you may abbreuiate the num¦bers which doe remaine, by 3. by 4. by 5. 6. 7. 8. 9. or by 10. for you must ab∣breuiate them as often as you can by any of the saide numbers, and it is to bée noted, that with whatsoeuer num∣ber of these, you doe abbreuiate the Numerator of your Fraction, by the same you muste abbreuiate likewyse the Denominator, so continuynge vntil they can no more bee abbreuied. And it is to bee vnderstande that if the Numerator and the Denominator

be euen numbers, as you maye know when the fyrste fygure is an euen nūber, or a 〈◊〉〈◊〉, thē maye you perceaue if both the Numerator and the Deno∣minator may be abbreuied by 10. by 8 by 4. or by 2. although y^t some times they maye bee abbreuied by three.

And if they be odde numbers, then muste you consider if they maye bee abbreuied by 9. by 7 by. 5. or by 3: but when the first number, as well of the Numerator, as of the Denomina∣tor are euen numbers, then may you well knowe that suche numbers maye bee abbreuied be 2. as is afore∣saide. And if you adde the fygures of the Numerator togither, in su∣che manner as you doe in makynge the proofe by nyne in whole Num∣bers: that is, if you fynde 9. 〈◊〉〈◊〉 ap∣peareth that you maye abb〈…〉〈…〉 that number by 9. And lykewise by 3. and sometimes by 6. if you fynde 6. it maye bee abbreuied by 6. and

alwaies by 3. if you finde 3. it is a signe that you may abreuiate by 3. And by whatsoeuer nomber that you doe a∣breuiate the numerator, by the same must you abreuiate likewise the de∣nominator, and if the first figures of the same nomber be. 5. or 0. you maye abreuiate them by 5. but if the firste fygures be both 0. they may be abre∣uied by 10. in cutting awaye the twoo Cyphers thus, as 〈◊〉〈◊〉 whiche maketh 2/••, & sometimes by 100. thus, as 〈◊〉〈◊〉 in cutting away the foure ciphers af∣ter this sorte, 〈◊〉〈◊〉 and then the 100/200 doe make ½, and after this maner haue I set here diuers examples, althoughe that all numbers cannot be abreuied by this rule, that is to saye, all those whiche maye bee well abreuyed by the fyrste rule afore∣sayde.

〈 math 〉〈 math 〉

3. Furthermore you shall vnder∣stande that sometimes it happeneth, that all the fygures of the numera∣tor are egall vnto them of the deno∣minator, which when it so happeneth, you maye thē take one of them of the numerator, and also one of them of the denominator, and it shall bee a∣breuyed as 555/888, beinge abreuiated af∣ter this maner commeth to ⅝. And yet it happeneth sometimes, that two, or manye fygures of the numerator are proportioned vnto two, or many fy∣guree

of their denominators and the other fygures of the same number doe beholde the one the other in thys proportion? Then may you take twoo or many fygures, as well of the nu∣merator as of the denominator, and by this maner the same number shall bee abbreuied, as 4747/〈…〉〈…〉 whiche beinge abbreuied by this rule, do come to 〈…〉〈…〉.

4. Also it happeneth somtimes that you woulde abbreuiate one number vnto the semblaunce or likenesse of another. And for to knowe if the same maye by abbreuied, and also by what number it maye bee abbreuied, you must diuide the numerator of the one number, by the numerator of the o∣ther, and likewise the denominator of the one, by the denominator of the other, for in case that after euery di∣uision there doe remaine 0. and that the twoo quotiens be 〈◊〉〈◊〉 all, then is one of them the number by the whiche the saide fraction must be abbreuied, as by exāple of 11••/〈…〉〈…〉. I woulde knowe if

they maye be abbreuied vnto 5/9, and for to doe this, you must diuide 115. by 5. and you must diuide 207. by 9. and there will come into bothe the quoti∣ents 23. by the which it appeareth that this number may be abbreuied by 23. 〈 math 〉〈 math 〉