The grounde of artes teaching the perfect vvorke and practise of arithmetike, both in whole nu[m]bers and fractions, after a more easie ane exact sort, than hitherto hath bene set forth. Made by M. Robert Recorde, D. in Physick, and afterwards augmented by M. Iohn Dee. And now lately diligently corrected, [and] beautified with some new rules and necessarie additions: and further endowed with a thirde part, of rules of practize, abridged into a briefer methode than hitherto hath bene published: with diverse such necessary rules, as are incident to the trade of merchandize. Whereunto are also added diuers tables [and] instructions ... By Iohn Mellis of Southwark, scholemaster.

NVMERATION.

BVt first to begin with the expressing of a Fraction, which is the numeration of it, you must vnderstand that a Fraction is repre∣sented by 2 numbers, set one ouer the other, and a line drawen betwéene them as thus, ⅓.••/4.⅘ 10/17 which foure Fractions you muste pro∣nounce thus: ••/3 one thirde parte: ¾▪ thrée quar∣ters: ⅖ two fifte partes: 10/17. tenne seauentéene partes.

Scho.

I vnderstande the forme of theyr ex∣pression and pronunciation, but their mea∣ning or valuation séemeth more obscure: yet I think that by the two first Fractions I vn∣derstand the valuation of the two later Fra∣ctions, and so consequentlie of other.

M.

Value them then, that I may perceyue your taking of them

Scholer.

⅖ betokeneth two fifte partes, that is to say, if one be diuided into 5 parts, that Fraction doeth expresse ij. of those fifthe

partes: 10/17 doth signifie, that if one be diuided into xvij. partes, I must take tenne of them. And this I gather of the two firste examples: for ⅓. that is one thirde parte, doth easily de∣clare, that if anye one thing be diuided into three partes, I muste take but one of them: so ¾ that is thrée quarters, doeth declare that one being diuided into four quarters, I must take (for this Fractiō) thrée of those quarters.

If there be no more difficultie in their Nu∣meration, thē I pray you go forward to their Addition and Subtraction,* 1.1 and so to the other kinds of workes for I vnderstande that the same kinds of workes be in Fractions, that be in whole numbers.

Maister.

There are the same kynds of workes in bothe, albeit the order of them is diuerse, as I will anone declare: but yet more in Numeration before we leaue it. You muste vnderstande, that those two numbers whiche expresse a Fraction, haue seuerall names.* 1.2 The ouermost whiche is aboue the line, is called the Numerator, and the other beneath the lyne, is called the De∣nominator.

Scholer.

And what is the reason of theyr diuerse names? For in mine opinion both bee

Numerators, séeing both they do expresse the numeration of the Fraction.

Maister.

You are deceiued: for one onelye (whiche is the ouermoste) doeth expresse the Numeration: and the denominatour doeth declare the number of partes into whiche the vnit is diuided, as in this example, when I say: Diuide a pounde weighte of Golde be∣twéene foure men, so that the firste man shal haue 2/15 the seconde 2/15 the thirde 4/15 and the fourth 6/15.

Now do you perceiue y^e by the denominator (whiche is one in al foure Fractions) it is in∣tended, that the pounde waight shoulde be di∣uided into so manye partes I meane 15, and by the foure seuerall numerators is limitted the diuerse portion that each man shold haue, that is, that whē the whole is parted into 15, the firste man shall haue 2 of those 15 partes: the second man thrée of them: the third man 4: and the fourth man 6. And so may you sée y^e seueral offices (as it were) of those two num∣bers, I meane of the Numerator and the de∣nominator.

And hereby you perceiue, that a man can haue no more parts of any thing than it was diuided into, nether yet aptlie so many: so that

it were vnaptly sayd: You shall haue 15/15. that is xv fiftéene partes of any thing, séeing it were better sayde: You shal haue the whole thing.

Sc.

So doth it appeare reasonablye: for the labour is vaine, to diuide anye thing, and than to applie the Diuision to no vse. And much lesse reasonable were it to say 16/15: for if the whole be diuided into 15 parts on∣ly, it is not possible to take 16 of them, that is to say, more than altogither.

Maister.

This is true touching the proper and apte vse of the name of a Fraction:* 1.3 yet improperlye, and after a vulgare acceptation (for easinesse in worke) both those formes be called Fractions, because they be writtē like fractions, although they be none in déede for 15/15, and generally all suche other: where the Numerator and Denominator be equal, are not Fractions: but the whole thing with all his partes. And so 16/12 is not to be called a fra∣ction, but a mixt number, of a whole number and a Fraction: for it is as muche, as 1 4/12, that is one whole one, and 4 twelue partes, as shall be declared in Reduction. Therefore they doe abuse the names, that

call them Fractions, where the Numerator is either equall or greater than the Deno∣minator.

Sc.

But is there any néedefull cause why they should so abuse the name?

Mai.

There is cause why they shal some∣times, for easinesse in worke, write some nū∣bers after that sorte, like fractions: but they néeded not to call them fractions, but as they be whole numbers or mixt numbers (that is whole numbers with Fractions) expressed like fractions.

Nowe must you vnderstande, that as no fraction properly can be greater than 1, so in smalnesse vnder one the nature of Fractions doeth extende infinitelye: as the nature of whole numbers is to increase aboue one infi∣nitelye, so that not onely one, may be diuided into infinite Fractions or parts, but also eue∣rye Fraction maye be diuided into infinite Fractions or partes, whiche commonlye bee called Fractions of Fractions, and they be expressed diuerslye: As for example, 3/••.⅔.½, that is thrée quarters of two third partes, of one halfe parte. Whereby is signifyed, that if one be diuided into two halfes, and the one halfe into thrée partes, and two of

those thrée partes, be diuided idyntlye into foure quarters, this Fraction of Fracti∣ons doeth represente three of those quar∣ters.

Scholer.

I praye you lette me prooue by an example in common money, whether I doe rightlye vnderstand you or no. One Crowne, which I take for an vnit, doth con∣taine 60 pennies, therefore the halfe of it is thirtye pence: ⅔ of that halfe is 20 pence, whereof 3/•• is fiftéene pence, so then 15 pence is 3/••.2/••.½ of a Crowne. And so 3 pence is ¾.⅔.½. of a shilling.

Mayster.

You perceyue this well y∣nough, but how happened that you founde no doubte in the forme of writing these Fra∣ctions, séeing the two latter Fractions haue no line betwéene their numbers, as the firste hath?

Sc.

Because I had forgotten (as Scho∣lers oft times doe) that that was tolde me before: but I praye you, expresse the reason thereof.

Mai.

This forme is but voluntarye, and therefore hath none other reason than the will of the diuiser, which forme many do fol∣low. Some other doe make lines betwéene e∣uery

Fraction, and adde wordes of distincti∣on, after this sorte, ¼ of ⅔, of ½, which forme is good also.

Some other expresse them thus 〈 math 〉〈 math 〉 in slope forme, to distincte them from seuerall Fractions of one whole number, for if they were set in one right line thus,* 1.4 ¾ ⅔ ½. then oughte it to be pronounced, thrée quarters, and two thirde partes and an halfe, which maketh al∣most two whole vnits, lacking but one xij. part. And so is it nothing agréeable with the other Fraction of Fractions, wherefore it is a great ouersight in certayne learned men, which doe expresse them so confusedly with such seuerall Fractions, that a man can not know the one from the other.

Therefore some men (as Stifelius) doe expresse without a line numbers of proporti∣on, being applied to Addition or Subtracti∣on: because they must be takē as two, where the line in Fractions maketh them to be ta∣ken for one: for of the Numeratour and De∣nominatour is made one number.

Scholer.

* 1.5Then I perceiue there be thrée seuerall varieties in Fractions: First when one only Fractiō is set for one nūber, as 4/••, y^t

is foure fifth parts. The second, is whē there be set two or more seuerall Fractions of one number, as ⅘ 2/5, that is iiij. ninth partes, and two fift parts. The third sort is Fractions of Fractions, as 4/9 2/5, that is 4 ninth parts of two fifth parts.

Ma.

You haue said well, if you vnderstand well your own wordes.

Scho.

If it shall please you, I will by an example in the parts of an old Englishe An∣gell expresse my meaning.

Mai.

Let me heare you.

Scholer.

The olde Englishe Angell did containe 7 shillings 6 d, that is 90 d. Nowe ⅘ of it, is 72 d. And of the same 90 pence, if I take ⅘ and ⅖, that is foure ninth partes, and 2 fifth partes, 4/9 is 40, and ⅖ is 36, which both make 76: but if I take 4/9 of ⅖, that is foure ninth partes of two fift partes, séeing ⅖ is but 36, then 4/9 of 36 will yield but 16: for 1/9 of 36, is but 4, and that taken foure times maketh 16.

Maister.

This is plainely expressed, and truely and hereby (I doubte not) but you doe perceiue, y^t as great a differēce as is betwéen 16 & 76, so much differēce is betwéene these two Fractions 4/9 and ⅖: and 4/9 of ⅖.

And now that you vnderstande these varie∣ties, I will procéede to the rest of the works: first admonishing you, that there is an other order to be followed in Fractions than there was in whole numbers, for in whole num∣bers this was the order: Numeration, Addi∣tion, Subtraction, Multiplication, Diuision, and Reduction, but in Fractions (to followe the same aptnesse in procéeding from the easi∣est workes to the harder) we must vse this order of the workes: Numeration, Multipli∣cation, Diuision, Reduction, Addition, and Subtraction.

Scho.

That Multiplication and Diuision shoulde goe togither, and Subtraction to followe Addition, naturall order doeth per∣swade: but why Multiplication shoulde be first in order here next to Numeration, and Reduction in the middle, I desire to vnder∣stand the reason

May.

As in the Arte of whole numbers order woulde reasonablye beginne with the easiest, and so goe forwarde by degrées to the hardest, euen so reason teacheth in Fractions the like order. And considering that Addition or Subtraction of Fractions can very seldom be wrought without multiplication and Re∣duction:

and contrariwayes, Multiplication and Reductiō may be wrought without this forme of Addition or Subtraction. Therfore was it orderly required, that Multiplication and Reduction shoulde goe before Addition and Subtraction. And the same reason ser∣ueth for the placing of Multiplication before Reduction.

Sch.

Then if Multiplication be the easi∣est, I pray you declare the forme of it first by rule, and then by example.

May.

Your example is good.

Notes

• * 1.1

  ☜
• * 1.2

  Numera∣tor, and Denomi∣nator.
• * 1.3

  ☜
• * 1.4

  ☞
• * 1.5

  Three se∣uerall va¦rieties.