NVMERATION.

NVMERATION is that A­rithmeticall skill, whereby we may duely value, expresse and reade anye number or summe propounded: or else in apte fi­gures and places, set down any number kno­wen or named.

Why? then me thinketh you put a difference vetwéene the value and the Fy­gures?

Yea so doe I: For the valewe is one thing, and the figures are an other thing: and that commeth partely by the diuersity of figures, but chieflye of the places wherein they be set.

Then I muste knowe there thrée thinges: the Value, the Figure, and the Place.

Euen so: but yet adde Order to them as the fourth. And firste marke, that there are but tenne figures, that are vsed in Arithmetike: and of those tenne, one doth sig­nifie nothing, whiche is made like an 0, and is called priuately a Cyphre, thoughe all the [Page] other sometime he likewise named. The o­ther nine are called Signifying figures, and be thus figured.

1 2 3 4 5 6 7 8 9 And this is their value. j.ij.iij.iiij.v.vj.vij.viij.ix.

But here muste you marke, that euery Fi­gure hathe two values: One alwayes cer­tain that it signifieth properly, which it hath of his forme: and the other vncertaine, which he taketh of his place.

A Place is called the seate or roome that a Figure standeth in. And looke howe manye Figures are written in one summe, so ma­nye places hathe that whole number. And the firste place muste bée called that that is nexte to the ryghte hande, and so recko­ning by order towardes the lefte hande, so that that place is laste, that is nexte to the lefte hande. As for example: If there stoode before you sixe men in a rowe, side by side, and you shoulde tell them as they stand in order, beginning wyth the man that were nexte to youre righte hande: then hée that were nexte hym shoulde bée called the se­conde, and so foorthe to the farthest from [Page] your ryghte hande, whyche is the sixt and the laste.

Syr, I perceiue you well: so might I recken letters or anye other thing. As if I shoulde write eight letters after this order, a, b, c, d, e, f, g, h, now muste I say h, is the first g the ij, f the iij, e the iiij, d the v, c the vj, b the vij, and a the viij.

That is wel done. And after the same sort vse hereafter, that what I declare by one example, doe you expresse by an other, and so I shall perceyue whether you vnder­stande it or no. And so passe ouer nothing, till you perceiue it well, and be experte therein.

Sir, I pray you howe manye of these places be there in all?

There is no certaine number of them, but they are sometimes more and som­times fewer, according to the summe that is expressed. For so many as the figures are, so many are the places: and the laste place is so called, not bycause it is laste of all other, but it is the laste of that present summe, and it maye bée the middle place in an other summe.

Me séemeth I perceiue this verye well, as touchyng the order of reckoning [Page] of the places: But as for the number of them, you say there is no certaintie. Nowe there resteth to declare the value of the figures by diuersity of places, which you called, the Va­lue vncertaine.

But firste let me heare whether you knowe perfectly the certaine value.

Yes sir, as you wrote them, so I marked them.

Howe write you then fiue?

By this figure 5.

And howe sixe?

Thus, 6.

Write these thrée numbers eache by it selfe as I speake them .vij.iiij.iij.

7.4.3.

How write you these foure other, ij, j, ix, viij?

Thus (I trow,) 2, 1, 6, 8.

Nay, there you misse: Looke on mine example againe.

Sir, truth it is, I was too blame, I tooke 6 for 9, but I will be warer héreaf­ter.

Nowe then take héede, these cer­taine valewes euerye figure representeth, when it is alone written withoute other Fy­gures [Page] ioyned to him. And also when it is in the firste place, though manye other doe fol­lowe: as for example: This figure 9 is ix. standing now alone.

Howe? is he alone and standeth in the middle of so many letters?

The letters are none of hys fel­lowes. For if you were in Fraunce in the middle of a M. Frenche men, if there were no English man with you, you woulde rec­ken your selfe to be alone.

So it is. Then 9 without more figures of Arithmetike, betokeneth ix, what­soeuer other letters be aboute it.

Euen so, and so dothe it, if it be in the firste place ioyned with other, how many soeuer do followe, as in this example, 3679. you sée 9 in the firste place, and doth betoken nine, as if he were alone.

I perceiue that. And dothe not 7 that standeth in the seconde place, betoke vij? and 6 in the third place, betoken vj? And so 3 in the fourth place, betoken thrée?

Their places be as you haue said, but their valewes are not so. For as in the firste place, euerye figure betokeneth hys owne value certaine onelye, so in the second [Page] place euery figure betokeneth hys owne va­lew certaine ten times: as in the example, 7 in the seconde place is seuen times x, that is, lxx. And in the third place, euery figure beto­keneth his owne valewe a hundreth times, so that 6 in that place betokneth vj.C. And in the fourth place, euery figure betokeneth his owne value a M. tymes, as in the foresaid number 3. in the fourth place, standeth for [...]. M. And in the fi [...]th place, euerye figure stan­deth for his owne valewe x.M. times. And in the vj place a C.M. times. And in y^e v [...]. place a M.M. times: And in y^e viij. place x.M.M. so y^t euerye place excéedeth the former x. times.

As thus: if I make th [...]s Number at al aduentures, 9 [...]359 [...]84, heere are eyght places. In the first place is 4, and betokeneth but foure: in the second place is 8, and betok­neth x. times 8, that is, 80: In the third place is 6, and betokeneth sixe hundreth: In the 4. fourth place 9 is nine thousand. And [...] in the fifth place is x.M. times 5, that is, fiftie M. So [...] in the sixte place, is a C.M. times [...], that is CCC.M. Them in the seuēth place, a M.M. And 9 in the eighte place tenne thousande thousande times 9, that is xc.M.M. But nowe I can not easily nor quicklye [Page] reade it in order.

That shall you practise by thys meanes. First putte a pric [...]e ouer the fourth figure, and so ouer the seauenth. And (if you haue so many) ouer the tenth, thirtéenth, six­teenth, and so forth, still leauing two figures betwéene eche two prickes. And those twoo roomes between y^e pricks, are called ternaries.

Then begin at the last pricke, and see how many figures are betwéene him and the end, which can not passe thrée, reconing hymselfe for one: then pronounce them as if they were written alone from the reste, and adde at the end of theyr valewe so many times thousand as your number hath prickes.

After that come to the nexte thrée figures, and sounde them as if they were aparte from the rest, and adde to their valewe so manye tunes thousandes, and there are prickes bée­twene them, and the first place of your whole number. And so do by euery other three Fi­gures following, if you haue moe. As in ex­ample, 91359684. this was your number.

Put a pricke ouer [...] in the fourth place, and ouer in the seauenth place, and then no more, (for your places come not to tenne) as thus: 91359684.

[Page]Nowe go to the last pricke ouer 1, and take it and the figure 9 that foloweth it, and value them alone.

91 that is xcj.

So is it: but then adde for the number of your prickes twice M.

That is xcj, thousande thousande.

So is it. Then take the thrée o­ther figures from one to the next pricke, and value them.

359. that is CCC.lix.

Now adde for the one pricke▪ that is betwéene them and the firste place, M.

CCC.lix. thousande.

Then come to the other thrée f [...] ­gures that remaine.

684. that is, vj.C.lxxxiiii.

Nowe haue you valued all. And at the end of the laste number you shall adde nothing, bicause there remayneth no pricke nor number after it: yet proue in an other number, as thus, 2 3 0 8 6 4 0 8 9 10 5 3 4 0.

2 3 0 8 6 4 0 8 9 1 0 5 3 4 0. I haue pricked them as you taught me: but I am in doubt, whether I haue done well or no, by­cause of the Cyphars: for I remember, you tolde me that they doe signifie nothing, and [Page] therefore I doubte whether I shoulde recken them for a figure in setting of the pricks: and againe, I know not wherefore they serue.

That wil I tel you now. Indéede they are of no value themselues, but they serue to make vppe number of places, and so maketh the figure following them to bée in a further place, and therefore to signify the more value: as in this example 90, the Ciphre is of no value, but yet he occupieth the fyrste place, and causeth 9 to be in the se­conde place, and so to signifie tenne times 9, that is, xc, so that two Ciphres thrusteth the figure following them, into the thirde place, and so forth.

Then I perceiue in the example aboue I haue pricked wel ynough: for tho gh that Ciphre that is pricked signifie nothing, yet must he haue the pricke, bicause he came in the xiij place. Then wil I proue to num­ber that sūme. First there is 230 M.M.M.M. and then foloweth 864 M.M.M. And what shall I nowe doe? There is a Cyphre in the third place, and no figure after him, but they that haue reckened.

He didde serue for them that you haue alreadye reckened, to make them in a [Page] place further than they shoulde be if he were away: and therefore nowe you shall let him goe. And so doe alwayes when he occupieth that place next before any pricke, whiche is the laste of that Ternarie, and a Cyphre in the laste place doth nothing.

Then shal I saye but 89 M.M.

So, but goe forth.

105 thousande. Nowe are all my prickes spent, and yet remaine 340, so that I muste value them CCC.xl. only.

Nowe can you recken after thys sorte: and remember, that euery such roume so parted, is called a Ternarie or Trinitie.

Some doe parte such great numbers with letters, after this manner.

2c 3b 0a 8b 6c 4a 0b 8c 9a 1b 0c 5a 3b 4c 0a. In whyche example yée maye sée, that a supplyeth the roume of your pricke. And some doe parte the numbers with lines after this forme. [...]. where you sée as many lines as you made prickes, and is one intente, saue that the lines doe more plainely parte euery thrée figures, according as they shoulde be valewed vnder one Deno­mination.

Yea sir, but if you shoulde shewe me a number so parted, I shoulde take it for many numbers, and not for one.

So might you doe, not knowing my meaning. But what if I didde set forth the number without lines, and your selfe (for the ease of reckning) did so part it with lines, woulde you forget wherefore ye did it, and then take them for many numbers?

No I trowe not, but yet I doubte.

Thou vse that that you like best, for all the thrée wayes are to one intent, saue (as I said) that the lines vs more plainely di­stinct the denominations.

What call you Denominations?

It is the laste value or name ad­ded to any summe. As when I saye: CC.xxii. pounds: pounds is the Denomination. And likewise in saying: as men, men is the Denomination, and so of other. But in this place (that I spake of before) the laste number of euery Ternarie, is the Deno­mination of it. As of the first Ternarie, the denomination is vnites, and of the seconde. Ternarie, the Denomination is thou­sandes: and of the thirde Ternarie, thousand [Page] thousandes, or millions: of the iiij. thousande thousand thousandes, or thousande Milli­ons: and soforth.

And what shall I call the value of the iij. figures that maye bée pronounced before the Denominators? as in saying: 203 000000, that is CCiij. millions. I perceyue by youre wordes, that millions is the De­nomination: but what shal I call the CCiij. ioyned before the Millions?

That is called the Numerator or valewer, and the whole summe that resul­teth of them bothe, is called the Summe, va­lue or number.

Nowe is there anye thing else to be learned in Numeration? or else haue I learned it sully?

I might here shew you who were the firste inuentors of this Arte, and the rea­sons of all these things that I haue taughte you, but that wil I reserue till ye haue lear­ned ouer all the practise of this Arte, leaste I shoulde trouble your witte, with ouer many things at the firste.

But yet this muste you marke, that there are thrée kindes of number: one called Di­gits, an other articles, and the thirde mixte [Page] numbers.

A Digit is any number vnder 10, as this: 1, 2, 3, 4, 5, 6, 7, 8, 9.

And 10 with al other that maye be diuided into tenne parts iust, and nothing remaine, are called Articles: suche are 10, 20, 30, 40, 50, &c. 100, 200, &c. 1000. &c.

And that number is called mixt, that con­taineth Articles, or at the leaste one article, and a digit: as 12, 16, 19, 21, 38, 107, 1005. and so forth. And for the more ease of vnderstan­ding and remembraunce marke this: The diget number is neuer written wyth more than one figure, but the article and the mixt number are euer wrytten wyth more than one figure. And thus they differ, that the ar­ticle hathe euermore this Cyphre c, in the firste place: and the mixte number hathe e­uer there some Diget.

By these laste wordes, I perceiue it muche better than I did before, and now (I thinke) I wil neuer misse to knowe those thrée asunder.

If you remember nowe all that I haue sayde, you haue learned sufficientlye this firste kinde of Arithmetike, called Nu­meration, Howbeit, I will yet exhorte you [Page] nowe, to remember bothe this that I haue saide, and all that I shall saye, and to exer­cise your selfe in the practise of it: For Rules without practise, are but a light knowledge: and practise it is, that maketh menne perfect and prompt in all things.

And as you haue learned to gather and ex­presse the value of a summe propounded, and set downe before you: so muste you practise to marke note, or write downe, with apte fi­gures, and in due places, any number, onely named or recited to you, or of your selfe ima­gined: as for a proofe: How note you, or write downe this summe, fiue thousande, two hun­dreth, fiftie and seuen.

This troubleth me nowe, whe­ther I shoulde beginne at the firste figure or at the last. For reason (me thinketh) should cause me to beginnne at the firste: and yet if I write it as you speake it, I muste beginne at the laste.

When you knowe youre places perfectly, you maye beginne where you list. But the more ease for your hande is to begin with the laste, that is to say, as I did speake them. Yet for the more suretie, a while you maye beginne with the firste, repeating my [Page] wordes backewarde thus: Seauen, Fiftie, two hundreth, fiue thousande: or else soun­ding them all by their diget or valewer, as thus: seauen, fiue, two, fiue: for that way is easiest. But then muste you looke wel, whe­ther there be any Ciphre in your summe that he may be sette in his place. As if youre last valewer of youre summe (as you speake it) be aboue 9, then is there a Cyphre in the first place. And if it be a hundred or aboue, then is there two Cyphres one in the first place, & an other in the seconde, and so forth.

But bicause this thing is such that cannot be set forth without manye wordes, I think beste here nowe at the ende of Numeration to adde a table easie and ready for the first ex­ercise of it.

Lo, this is the Table. [Page]

The [...]

This Table (as you may sée) hathe eleuen places, and in eche of them are sette al the di­gites, whose certaine value is written in the right hande of the Table, & the value vncer­taine [Page] on the left hande. So that by this table you may learne bothe howe to expresse anye number that you liste, (if that it excéede not eleauen places) that is to say, lxxxx. thousand Millions, and so maye you by the helpe of it, value all summes proposed vnder the sayde number.

For example: take the summe that I pro­posed before, which was fiue thousand, two hundred, fiftie and seauen. And if you wil ex­presse it, take the firste number (as I speake it) which is fiue M. whose valuer or certaine value is v. and his vncertaine value or de­nomination is M. First you shall séeke at the right hand of the valuer v. Then séeke along vnder the tytle of Denomination towarde the left hande, til you find thousands, and vn­der it right at the soote of the Table, is the number of the place, that is the fourthe, wherein you muste write youre diget or va­luer fiue.

Afterwarde come to the seconde parte of the number, two hundred, whose valuer is 2, and his denomination C. Séeke two at the right hande of the Table, and goe along vnder the denominations towarde the lefte hande, til you come vnder C: then looke to [Page] the table, and there shall you sée the number of the place, that is to say, thrée, wherein you muste set your diget 2.

Then doe so by your other two numbers that remaine, and you shall finde fiue in the second place for your fiftie, and 7 in the first place for your seauen. And thus maye you doe with other numbers.

Maister I thanke you hartilye. I perceiue you séeke to instruct me most plain­ly and brieflye, and not to hide youre know­ledge with subtile wordes as many doe. For this rule is so plaine, that I can desire it no plainer. And though it seeme somwhat long, yet I perceiue it to be a sure way.

So is it, and thoughe it be long, yet it is neyther too long, neyther too plaine for yong learners that lacke practise: for this table is in steade of a teacher, to them that lacke one. But nowe I truste I haue said y­nough of Numeration: which after you haue wel practised, then may you learne forth.

Yet I praye you in one thing to tel me your iudgement. Why do men recken the order of the places backewarde, from the right hand to the lefte?

In that thing all men doe agrée, [Page] that the Chaldeyes,☜ which first inuented this Arte did set these figures as they set all their letters: for they wryte backewarde as you terme it, and so do they reade. And that may appeare in all Hebrue, Chaldeye, and Ara­bike bookes, for they bée not only written frō the right hande to the lefte, and so muste bée read, but also the right end of the booke is the beginning of it: whereas the Gréekes La­tines, and all nations of Europe, do write & reade from the left hande towarde the right: And all their bookes begin at the left side.

That reason doth satisfie me.

It neyther sact [...]fieth mée, neyther lyketh me well, bycause I sée that the Chal­deys and Hebrues doe not so vse their owne numbers, as at another tyme I will declare. But this playne reason may best satisfie you presently: That séeing in pronouncing of numbers we kéep the order of our owne rea­ding, from the lefte hande to the right: And againe, we doe euer name the greater num­bers before y^e smaller: it was reason, that the lesser places conteyning the lesser numbers, should be set on the right hand, and the grea­ter places conteining the greater numbers, to procéede toward the left hande.

This reason is to me so plaine, that it séemeth now againste reason to make a doubte of that order. So that nowe for Numeration I am satisfied: so that onelye practise shal make me fully ready and expert in it. And in the meane season, I desire to learne the other kindes of Arithmetike.

That is well saide: but what should you next learne can you tel?

I remember you said that Addi­tion was nexte.

Euen so, and what that is muste you firste knowe,

ADDITION.

ADditiō is the gathering to­gither and bringing of twoo numbers or more into one totall summe: as if I haue 160 Bookes in the Latyne tong, and 136 in the Gréeke tongue, and would know how many they be in all, I muste write these two numbers one ouer an other, writing the greatest number highest, so that the first figure of the one, bée vnder the firste figure of the other. And the [Page] seconde vnder the seconde, and so forth in or­der.

When you haue so done, draw vnder them a right line, then wil they stand thus. [...] Nowe beginne at the firste places, toward the right hande alwayes, and putte togither the two first figures of those two numbers, and looke what commeth of them, write vnder them, right vn­der [...] the line. As in saying, 6 and 0, is 6. Write 6 vnder 6: as thus.

And then go to the second figures, [...] and doe likewise: as in saying, 3 and 6 is 9: write 9 vnder 6 and 3, as here you sée.

And like wise do you with the fi­gures [...] that be in the third place, say­ing: 1 and 1 be 2: write [...] vnder thē, and then will your whole summe appeare thus.

So that now you sée, that 160, and 136 doe make in all, 296.

What? this is very easie to doe, me thinketh I can do it euen sith.

There came thorough Cheapeside twoo droues of cattell: in the first was 848 shéepe, and in the second was 186 other beastes.

[Page]Those two summes I muste [...] write as you taughte me, thus.

Then if I put the two first fi­gures [...] togither, saying: 6 and 8 they make 14. That muste I write vnder 6 and 8, thus.

Not so, and here are you twice deceiued. First, in going about to adde togy­ther two summes of sundrie things, whiche you ought not to doe, excepte you séeke onelis the number of them, & care not for the things. For the summe that should resulte of that ad­dition, should be a summe neither of shéepe, nor other beastes, but a confused summe of both. Howbeit sometimes yée shal haue sum­mes of diuers denominations to be added, of which I will tell you anone: but first I will shew you, where you were deceiued in an o­ther pointe, and that was in writing 14, (which came of 6 and 8) vnder 6 and 8, which is vnpossible. For, howe can twoo figures of twoo places be written vnder one figure, and one place?

Truth it is: but yet I did so vn­derstand you.

I saide indéede, that you shoulde write that vnder them, that did resulte of thē [Page] both togither: which saying is alwayes true, if that summe do not excéede a Digit. But if it be a mixte number, then muste you write the Digit of it vnder your figures, as I haue said before: but and if it be an Article, then write 0 vnder them, and in both sortes you shall kepe the article in your mind. And ther­fore when you haue added your seconde fi­gures, which occupy the place of tennes, you shall put that 1 therto, which you kept in your minde: for though it were ten indéede, yet in that place it is but as one, because, that euerie 1 of that place, is ten, for it is the place of tens. And in like manner: if you haue in the se­cond place so great a number, that it amoun­teth aboue 9, then write the digite, and re­serue the article in your mind, euer adding it to the next place following: and so of all o­ther places, how many so euer you haue. And if you haue a mixt number, when you haue added your last figures, then write the digit vnder the last figures, and the article in the next place beyond them: so shall your number resulting of Addition, haue one place more than the numbers which you should adde to­gither.

Now doe I perceiue you, and the [Page] reason of this is, (as I vnderstande) because that no one place can contain aboue 9, which is the greatest figure that is, and then all tens or articles must be put to the next place follo­wing: for euerie place (as I may sée) excée­deth the other place next before him, by 10.

Now (if it please you) I will returne to my example of Cattell. But I remember you saide, I might not adde summes of sundry things togither, and that might I sée by rea­son.

Trueth it is, if you séeke the due summe of anie things, but if you onelie seeke a bare summe, & haue no respect to the thing, then were it better to name the summe onely without anie thing, as in saying 848, wyth­out naming shepe, or any thing else. And like­wise 186, naming nothing.

Now let me see: howe can you adde those two summes?

I must first set them so, that the two first figures stand one ouer an other, and the other each one ouer his fellow of the same place: then shall I drawe a line vnder them both. And so likewise of other figures, set­ting alwayes the greatest number highest, thus, as foloweth:

[Page]Then must I adde 6 to 8, which [...] maketh 14, that is mixte number: therefore muste I take the diget which is 4, and write it vnder 6 & 8 kéeping y^e article 1 in my mind thus.

Nexte that doe I come to the se­cond figures, adding them togither, saying, 8 and 4, make 12, to which I put the 1 reserued in my minde, and that maketh 13, of which number I write the diget 3 vnder 8, and 4, & kepe the article in my mind thus: [...] Then come I to the thirde figures, saying: 1 and 8, [...] make 9, and 1 in my minde maketh 10. Sir, shall I write the cipher vnder 1 and 8?

Yea.

Then of 10 I write the cipher vn­der 1 and 8, and kéepe the article in my mind.

What néedeth that, séeing there followeth no more figures?

Sir, I had forgotten, but I will remember better hereafter. Thē séeing I am come to the last figures, I muste write the cipher vnder them, and the article [...] in a further place after the cipher, thus:

So now ye sée, that of 848, and [...]86 added together, there amounteth 1034.

Nowe I thinke I am perfite in Addition.

That wil I proue by this example.

There are two armies of souldiours: in the one are 106800, and in the other 9400: How many are there in both armies say you?

Firste I sette them one ouer an other, beginning with the firste [...] numbers at the right hād, thus. But the neather nūber wil not match the ouer number.

That forceth not.

Then doe I adde 0 [...] to 0, and there amounteth 0, that must I write vnder y^e first place, thus.

Well saide.

Then likewise in the seconde place I adde 0 to 0, and there [...] ariseth 0, which I write vnder the second place, thus.

Then I come to the thirde place saying: 4 and 8 make 12, of whiche I write the diget 2, and kéepe the [...] article 1 in my minde, thus.

[Page]Then adde I 9 to 6, whiche [...] maketh 15, to that I adde the article 1 that was in my mind, and it is 16. I write 6 vnder 6 and 9, and keepe one in my minde, thus.

Why doe you not write bothe figures, séeing you are come to the last couple of numbers?

Nay reason sheweth me,☜ that I must adde that article that is in minde, vn­to the next figure of the ouer summe, thoughe there be no more in the neather summe.

That is well considered: then doe so.

Then say I, 0 in the ouer summe, and in my minde, maketh 1, that I write vn­der 0: Then foloweth there yet one more in the ouer summe, which hath none to be added to it, for there is none in the neather summe, nor yet in my mind, therfore I thinke I must write that euen as it is.

Yea.

Then doth my whole summe ap­peare, [...] thus.

If you marke this, you haue learned perfectly the cōmon addition of all summes [Page] which are of one denomination: so that ye ob­serue this also, that in Additiō you must haue two numbers at the least, or else how cā you say that you do adde? And euer let the greatest number be writtē highest, for that is the best way, though it be not necessarie.

And forget not this, that if you haue many numbers to adde togither, you shall haue of­tentimes an article of a greater value thā 10: sometimes 20, sometimes 30, sometimes more, yea, peraduenture 100. Therefore, as you did with the article 10, so do with them, reseruing them in your minde, and adding to the number next following, so many as their valuer or value certaine is: that is to say, 2 for 20, 3 for 30, and so forth of other. But if the article be 100,☞ then must you not adde the ar­ticle to the next figures following, but to the third figures from them, as I will shewe you anone by example. And if it chaunce the number to be such, that it doe comprehende twoo sundrie articles, (that is, one of tennes, and an other of hundreds) then muste you reserue them both in your minde, and adde the article of tennes, to the figures that fo­low next, and the article of hundreds, to the figure of the third place from thence.

[Page]Nowe take this example for [...] all. I would ad these xiij. sūmes in one, whiche I set after thys maner. Then do I beginne and gather the summe of the first fi­gures, whiche commeth to 107. For first I take 9. ther x. times, and that is 90: then 9 and 8 is 17, that is in all 107. of whiche summe I write the 7 vnder the firste figures, and then haue I an article of an hundred in my minde, whyche eyther I must kéepe in my minde till I come to the thirde figures, which are in the roomes of hundreds, or else I maye for feare of forgetting, write this one (béeing of the third place in your of come) vnder the third rowe of figures, making two lines, as you see here done. And then must I write the digites vnder the lowest line: and this is the surest way, when the summe is so great, that the addition of one row passeth 100.

When I haue so done, I must then come to the seconde rowe of figures, and adde them togither, whiche doeth make 115. of whyche [Page] summe I write the dygitte 5 [...] vnder the same seconde row, and then I haue a mixt num­ber remaining of two figu­res, of which the 1 (that stan­deth for 10) must be added to the second or next place after them that I did last adde. And the other y^t standeth for 100) muste be added to the thirde place from thence.

☞That is to say, the fourth place from y^e first line or row of figures.

Euen so. And thē wil the sum appeare thus. Then adde the third row of figures,☞ with the two vnities betwéene the line, and the summe amoūteth to 50: of which I write the Cypher vnder the same thyrde rowe, and the 5 vnder the nexte figures to­ward the lefte hand. And with my pen I giue a dashe to the two vnities betwéene y^e lines, whose valew I haue already added vnder the lowest line.

Then I adde the figures of the fourth row, with the 1 and 5 that are vnder them betwéen [Page] the two lines, & they make 29: [...] then dashe I the [...], & the 5, with my penne, as I did before the two vnities: & so write vnder y^e lowest line the 9 (y^t is the digit) vnder the fourth place: & the 2, that is the article, beyonde it, toward the left hand. So those summes doe make 29057.

This seemeth some­what harde, by the reason of so many nūbers togither. How­beit I think if I do often proue euē with this same example I shal be able to do so shortly, w^c any other summe.

So shall you. For it is often practise that maketh a man quicke and ripe in all things: But bycause of suche great summes there may chaunce to be some errour. I will teach you howe you shal proue whether you haue done wel or no.

That were a great helpe and ease.

Begin first with the highest num­ber, and then to all the other orderlie, & adde them togither, not hauing regarde to theyr places, but as though they were all vnities: [Page] and still as your number encreaseth aboue 9, cast away 9. Then goe forth, euer casting a­way 9 as often as it amounteth thereto: and so do till you haue gone ouer all the numbers that you intended first to adde, and whatsoe­uer remaineth after such addition and casting awaie of 9, write it in some void place by the ende of a line for the better remembraunce: & then putte togither the figures that result of the Addition, still casting away 9 also. And then that that remayneth, write at the other ende of that line: and if those two figures be like, then haue you well done by likelihoode: but if they be vnlike, then haue you missed. As for example in this present summe: The firste figure of the ouer line is 9, lette him go: then 8 and 8 is 16, take awaye 9, and there remayneth 7, adde to it 4 that followeth, and that maketh 11. from whiche if you take 9, there resteth 2: then come to the next row, whose first and seconde number are 9, there­ore ouerpasse them both, and take the 5 to [...]che 2 which did remaine in the first row, that maketh 7, putte thereto the 4 folowing, that maketh 11, thence take 9, and there remay­neth 2 [...]: nexte that, goe to the thirde line, whose two firste numbers you maye lette [Page] passe, bycause they are nines: then take the twoo whiche with the other two that remay­ned in the seconde rowe, make 6: then goe to the fourth rowe, whose two firste numbers let goe, and take the 6, to the [...] that remay­ned, and that maketh [...], take awaie 9, and there resteth 3, which with the 3, that is next, maketh 6. And so goe throughe all the other numbers, and you shall finde that there re­maineth 5, after you haue cast away 9 as of­ten as you find it: therfore write 5 at one end of a line in a voide place thus. [...]

Then gather all the figures of the totall summe which is vnder the lowest line, and cast away 9 as often as you finde it, as thus: seauen and 5 make 12, take awaie 9, and there resteth [...], to that if you adde the 2 that is laste (for you may let goe the 9) then doth it make 5, whiche you must write at the other ende of the line that you made in the voide place, and it wil be thus. [...]

And then you see that those two figures bée like, whereby you maye knowe that you haue done well, and so maye you proue in a­nie other.

If it please you, I wil proue in an other summe.

With a good will.

Then wil I take one of your for­mer examples, which was this.

First in the highest line, 8 and 6 make 14, then 9 taken away, there re­maine [...] 5, to which I adde the 1 that foloweth, and that ma­keth 6. Then come I to the second line, where I finde first 4 which with 6, maketh 10, from that I take 9, and there resteth 1, the next figure is 9, and therefore I let him alone, so find I one remaining, which I set at the ende of a line thus. [...] Then I come to the totall summe, and there I finde that al the figures put together make 10, from which I take 9 and there resseth 1 al­so, which I put at the other ende of the line thus, [...]

And because they be like, I know that I haue wel added.

So you knowe nowe both howe to adde two sūmes or more together: and al­so how to proue whether you haue done well or no:Addition of nūbers of diuerse denomi­nations. which thing also you may doe best by Subtraction. But because you cannot yet skil of it, I will let that passe till anone, and will teache you now how to adde sūmes of diuers [Page] denominations: whiche thing can neuer be but whē the one denomination is such that it conteineth the other certain times. And yet you shal adde them to the other, not after this sort as you did them that were of one deno­mination, but after such a sort as I will now shew you, that is to say.

If you haue a summe of diuers denominati­ons, then loke that ye set euery denomination by himself, with some note or figure of his de­nomination, as they be wont to be written. Then write your other sūmes so vnder that first, that euery one be set vnder the other of the same denominations, as for example: if your denominations be poundes, shillings, & pens, write pounds vnder poundes, shillings vnder shillings, and pens vnder pens, and not shillings vnder pens, nor pens vnder pounds.

Nowe that you haue spoken it, me thinketh it needeth not to warne me of it, for it were against reasō so to cōfound sūmes: but yet if you had not spoken of it, peraduen­ure I should haue bene deceiued in it.

If you doe say it is so plaine, I will speake no more of it, but with an exam­ple make the matter to appeare euidently.

[Page]Firste, one man oweth me 22 lb, 6 s, 8 d. An other oweth mée 5 lb, 16 s, 6 d. And an other oweth mée 4 lb, 3 s, I would know what this is altoge­ther. [...] Therfore must I first set down my greatest sum & thē the other, euery one vnder his denomination gréeing to the greatest summe, as here you see.

Then muste I beginne at smallest num­bers, (whiche must alwayes be set nexte the right hand) and adde them togither, and if the summe of them will make one of the next denomination, then muste I kéepe it in my minde till I come to that place, or else for more easinesse write it vnder that place be­twéene the double line, and vnder that place must I note the residue, if there remaine a­nie of the same denomination, but if there remaine none, then néede I to write vnder it nothing. And this is all that you muste marke in this Addition: for all other things are like to the other manner of Addition be­fore mentioned.☞ Therefore the chiefest point of this Addition is, to knowe the valewes of common coynes and rated summes. As how [Page] many shillinges be in a pounde: how many pence in a shilling, of which and of other like things, I will instruct you hereafter, in tea­ching of Reduction: But now I maye not disturb your wit from the thing that we are about.

Therefore let vs returne to [...] that former example, which I proposed of thrée detters, whi­che summes when I had sette orderly they stoode thus, with a double line vnder them.

Then to adde them vnto one sum, I must begin at the right hand, where the smallest denomination is, and adde thē togither first, saying: 6 and 8 make 14. Nowe séeing these 14 are pennies, and that 1 [...] pence make one shilling, which is the next [...] valewer, I take awaye 12 from 14, and there resteth 2, which I write vnder the pennies, and for the other 12, which maketh 1 shilling I write vnder the tytle of shillings, thus:

Then do I adde all the shillings togither, and finde them 25, to whiche I adde that 1, [Page] betwéene the two lynes, [...] that maketh 26, but bi­cause that 20 shillinges do make 1 pound, I take away 20 from 26, and for that 20 I write 1 vnder the pounds betwéene the two lines, and the other 6 that remayneth, I write vnder the shillings, as appeareth in the exāple before.

Then come I to the pounds, adding them all togither, and finde them to be 31: thereto I adde the 1 betwéene the [...] two lines, & that maketh 32, whiche sūme I write downe whole, because there resteth no greater denomination, and then my whole sūme appea­reth thus.

So is my totall summe, 32 lb. 6 s. 2 d. And this maye you prooue in an other lyke summe.

Then will I caste the whole charge of one moneths commons at Oxforde wyth batteling also.

Goe to, let me sée how you can doe.

One wéekes commons was 11 d. ob. q and my batling that wéeke was 2 d. q. q. The secōd wéekes commons was 12 d. and my batling 3 d. The third wéekes commons 10 d. ob. and my batling 2 d. q. c. The fourth wéekes commons 11 d, q. and my batling 1 d. ob. [...]. [...] These eyghte summes woulde I adde into one whole summe, and therefore I will sette them one ouer another, thus.

But I had forgot­ten, I shoulde haue set the greatest summe highest.

So is it commonly best, howe­beit, here it forceth not: and in such summes as this is, that go by order of wéekes, dayes, or yeares, it is better to kéepe that order, than to alter them, and to sette the greatest number highest, for that serueth for such summes as go not by order.

Then if I haue set them well ynough, I will begin to ad them thus.

[Page]Firste of the smallest [...] valewers at the right hande, which are called cées, I find [...], and séeing that 2 cées, do make one q, I wil write nothing vnder the cées, but will write 1 q for 2 cées, vn­der the kewes betwéen the lynes, as the exam­ple sheweth.

Then come I to y^e next valewers, where I finde 2 q, and to [...] them I adde the q that is betwéene the lines, and so are they 3 q: but because 2 q, maketh one q, I write one q vnder the farthynges be­twéene the lines, and the q that remaineth must I write beneth the nethermoste line vnder the kewes, thus.

Then come I to the farthinges, where I [Page] finde 3, and the other q that is betwéene the lines, maketh 4 farthings. And bicause 4 q make iust 1 pennie, I shall write nothing vnder the farthings, but must write 1 vnder the pens, betwéene the lines.

Next that must I adde the halfe pence to­gither, of which there are 3. but séeing that 2. ob. make 1 d, I must write 1 vnder the pens betwene the lines: but how shal I do it, for there is 1 alreadie?

Haue you forgotten how I didde in addition of the great summe before?☜ you must set it vnder the other, so shal they bothe stande for 2. For if you shoulde set it before or behinde the other, they should make 11.

I remember it nowe, and I per­ceiue the reason. Then I wil write 1 ob, vn­der the halfepence, and for the other two halfepens, which make 1 d, I write 1 vnder the pens: Then come I to the pens, & finde, that there are of them 52. then put I to them the 2 betwene the lines, and that maketh 54, which amoūteth to 4. s. 6 d: the 6 d I must write vnder the pens, and the 4 s. I must set (I suppose) farther toward the left hande by themselues.

Euen so.

Then ap­peareth [...] all my addi­tion thus. And the summe is 4 s 6 d. ob. q.

Now haue you done this well. But tell me, why did you writ kewe, cée, thus, q, c. & not rather thus q^c, as the fashion is?

Because I thoughte it was the best way for due ga­thering of euery denomination by himselfe.

So was it in déede. Wel now, can you tell how to proue this addition, and such other like of diuers denominatiōs, and to trie whether you haue done well or no?

I would I could.

That shall you doe by this meanes. First as you did begin to ad so rec­ken agayne euery denomination by it selfe, and when you finde so many small that doe make any other denomination, let them go, and kéepe in minde only the residue that wil make no greater denomination, and looke [Page] whether there be any such like value vnder the nether lyne, and if there bée, you haue well done, and so goe from one denominati­on to an other, vnto the ende.

But here must you note, that in gathering of the summes, ye must recken those figures that are written betwéene the lynes, with them that are written aboue them: as for an example, I will examine the summe that I did last adde, which stoode [...] thus.

Firste I finde 6 and 8, whiche maketh 14, from whiche I take 12, bicause it maketh one of the next denomination, and there remayneth 2, and vnder that place I sée a like figure, therefore I knowe that well to be done. Then come I to the s, where I find 1, 3, 16, and 6, that maketh 26, I cast away 20, for they make another denomination, that is to saye poundes: and the 6 which remai­neth, is lyke to the 6 that is written vnder them beneathe the lowest lyne, there­fore▪ that is well done also. And thence I goe to poundes, where I finde 1, 4, 5, 22, that is 32, to whiche summe agréeth another lyke [Page] vnder it. Therefore I iudge all well done.

I perceiue reason in this proba­tion. Nowe will I attempte the same in the summe that I did adde, whiche when I had ended adding, stoode as you maye sée in the example following.

Firste amongest the cées I find but two, which make one q euen, therfore there must nothing bee vnder the line for them: And amongste the kewes [...] I fynde 3, of whiche two make 1 q, there­fore I let them goe, and the one q, that is lefte, hath an o­ther lyke vnder his place, therefore that is well done.

Then the Far­thinges are iuste 4, which make 1 d. and therefore I let them goe. Amongst the halfe pence there is one od (for 2. must I cast away, bycause they made one penny) and vnto it answereth a like sum vnder it. The pens are 54. frō which I take away 48, that makes 4, s, and the 6 remay­ning [Page] agrée to a like figure set vnder thē. And last of all remayneth the 4 s, which the abie­cted pens did make: so I perceiue that I haue wel done. Now this will I not forget. But will this examination serue in all addi­tion?

It serueth for all addition of sun­drie denominations,☜ if the addition be made with two lines, (as were these) else it wil not serue, bycause that those summes whyche are here added betwéene the lines, in Addition by one line, are vnderstanded and not written: but I let that way passe, bicause as it is com­mō, so is it more deceiueable than this way, namelye if a mans memorie be either dull or troubled.

Yet it were good to knowe that way also.

If you desire to knowe it,Another forme of Addition. this it is in fewe wordes. Doe euerie thing as you did in this sorte of Addittion, saue that where you made here two lines, you shall make there but one: and those summes that you did here write betwéene the lynes, you must kepe in your memorie, and vse them (as you dydde here) each one when you come to his place.

Then they differ not, but in this, [Page] that this addition with two lines leaueth no­thing to memorie, but writeth downe all: and the other way committeth certaine numbers to memorie, as you taughte me in the first ex­amples of addition of small summes of one denomination. But what if a man vse it (as you say men do commonly) how shal it be ex­amined?

Séeing you are so desirous of it, I will shewe both an example of the additi­on, and also the manner to examine it.

I propose these thrée summes [...] to be added, and I gather firste the pence, as I did in the other sort, and I finde of them 8, 3, 9, that is 20, of which summe I bate away 12, whiche make 1 s, and kepe that 1 in my minde, and the rest, that is 8, I write vnder the pence.

Then do I adde the shillings togither, and finde of them 6, 7, 8, that is 21, whereof I bate 20, that make 1 lb, whiche I kéepe in minde, and to the other 1 that remaineth, I adde that one that came of y^e pens & was in my mind, whiche make 2, and them I write vnder the shillings.

Then doe I recken the pounds togither, 3, [Page] 6, 12, that is 21, and to them I adde the 1 in my mind that remayneth of [...] the shillings, which make 22, them doe I write vnder the poundes, and then my summe totall appeareth to be 22 lb, 2 s, 8 d.

Now to examine this sum and all suche like, you shall doe thus. Firste beginne at the lefte hande with the poundes,Another forme of proofe. and take from them that are aboue the lyne, 9, as often as you can: then that that remay­neth shall you double, and ioyne it with the shillings, and take awaye 9 from that as of­ten as you can, and whatsoeuer remayneth, yée shall take for it thrée tymes so muche, and putte to the pence: then take from all that summe 9, as often as you can, and what so remayneth after you haue wythdrawen 9 as often as you can, write that at the ende of a line, as I taughte you in the other Addi­tion.

And then come to the sum vnder the lire, beginning with the pounds, and doe euen as you did with the summes aboue the line, till you come to your pennies: and if the figure of [Page] the summe that remayneth after casting a­waye 9, (as often as you can) doe agrée wyth the other that remayned before of the other summe, whyche you did write at the ende of the line, then haue you done well, else not: and for an example, I will examine that laste summe which was thus:

First I shal begin at the [...] left hād with the pounds, putting them togither, whiche make 21, in which summe I finde 9 twice, (for twice 9 is 18) that I deducte, and there remayneth 3: that 3 must I double (as I saide) bicause it is the remay­ner of the poundes, and it will be 6. Then ga­ther I the summe of the shillings, whiche is 21, to the whiche I adde the foresaide 6, and then it is 27, wherein I finde 9 thrée times, and there remayneth nothing. This remay­ner should I take thrée times, but thrée times nothing, is nothing: therefore in this place is there nothing left to be added to the pennies. Wherefore I muste take the summe of penies alone, whiche is 20, from thence if I take 9 twice, there remayneth but 2, whych I putte vnto the ende of a lyne [Page] thus. [...]

Then I come to the poundes of the vnder number or totall summe, and there I finds 22, from whiche I take awaye 9 twice, and there remayneth 4: that 4 I double, and it is 8, then doe I adde that 8 to the shillings, and it maketh 10, from which I withdrawe 9, and there resteth one: then doe I take that 1 thráe times, and it maketh 3, whyche I adde to the 8 d. and it maketh 11, frō whi­che if I bate 9, there resteth 2, which is equal to the number noted at the ende of the line: and therby I perceiue that I haue done well.

But I doe not sée the reason of this.

No?The reason of this proofe. no more doe you of manye things else, but hereafter will I shewe you the reasons of all Arithmeticall operations: for this I iudge to be the beste trade of tea­ching, first by some briefe precepts to instruct a learner somewhat in the vse of the Arte,The best trade of teaching. be­fore he learne the reasons of the Art, and then may you afterward more sooner make him to perceiue the reasons: for harde it is to occu­pie a young learned witte with both the arte and the reasons of it all at once: howbeit hée [Page] shall neuer be cunning in déede in an arte, that knoweth not the reason of euerie thing touching it. But for this worke, bycause the reason is easie, I wil shew it you nowe. You knowe that if one pounde do remaine, it be­ing tourned into shillings, woulde make 20 s, in whiche number there is 9 contained twice, and 2 s beside. And therefore for one pounde you shal take 2 s, and so for euerie one pound 2 s.

I sée it well, for if there remained 7 lb, after the nines were cast away, I must take 14 s for that 7 lb. And so haue I caste a­way 14 times 9 s, and yet remayneth of eue­rie pound 2 s which maketh 14 s.

Like wayes in shillings, whiche containe 12 d: for euerie shilling, if you abate 9 pence there resteth 3 pence.

It is plaine ynoughe. And so if [...] shillings doe remaine, I muste take for it 15 d, that is thrée pence for euerie shilling, and yet in that so doing, I haue caste awaye fiue times nine pence.

Other workes haue as good rea­son, but I wil not stande aboute yéelding rea­sons now.

Yet one thing more I praye you [Page] shewe me, why did you write your number that remayned (after you hadde withdrawen all the nines) at the ende of a line? for I sawe no reason why that line did serue.

Did you euer marke a Carpen­ter when he wrought?

Yea many times.

And haue you not séene him when he hath taken measure of a boarde, that hée hath pricked it, and hath with a twitch of his hande drawen a line from the pricke that hée made?

Yes I haue marked that and haue séene some marke 3 or 4 lines by the pricke, some also haue I séene make a crosse by it, but that I perceiued was for the easie finding of their pricke.

And euen so is this line for the ea­sie finding of your remainer, and therefore some doe make a crosse, thus.

And set the one remainer aboue the crosse, and the other vnder the nether part of the crosse, as if I should set my two remainers thus. [...]

But there is another sort of proofe of Addition;Another kind of proofe most vsu­all and ap­test of all. to whiche the crosse serueth more méeter: & that is whe [Page] the addition is of diuerse denominations: and I would examine euerie denomination by it selfe, which waye though it be not much vn­like to the first proofe that I brought of suche diuerse summes, yet will I declare it, leasts you shoulde thinke that I would hide it from you.

You muste make so manie lines in your crosse, as you haue sundrie denominations: as if you haue but two denominations, then you may make it thus, that the ouer

part and the nether part may serue for one denomination, and the two sides for the other. And if you haue thrée de­nominations, as poundes, shillings, and pen­nies, then muste you make thrée

lines thus. The vpright line may serue for poundes, and the highest thwart line for shillings, and the lowest for pens: as for example I will take a summe thus added.

[...]

[Page]For the proofe of the which, bycause it con­tayneth thrée denominations, I must make a crosse of thrée lines, as in the page afore. Thē I recken first at the righte hand the pennies: 7, 1, 5, make 13, from whiche I take 12 for the next denomination, that is to saie, a shilling, and there resteth 1, which I muste write at one end of the neather thwart line.

After that I gather the summe of the shyl­lings, 2, 8, 12, which maketh 22. to them I put one that I toke of the pennies, and that ma­keth 23: from those I take [...]0, the quantitie of the next greater denomination, that is to say, a pound, and there resteth 3, which I write at the ende of the highest thwart line.

Thirdlie, I adde togither the pounds, 9, 12, 16, whiche make 37, to them I adde the 1 that came of the shillings, and then there is 38, wherein I finde 4 times 9, and 2 ouer, that 2 I write on the vpright line.

That doone, I come to the totall summe, and examine it, beginning at the pennies, where I finde but one, and cannot take 9 frō him, therfore I set him at the other ende of y^e neather thwart line: Then I come to the shillings, where I finde only 3, which bicause it is lesse than 9, I sette it at the other ende [Page] of the line of shillings, that is, the ouermoste thwart line.

Last of all, of the 38 lb, I take foure times 9, which is 36, and there remayneth 2, which I write vnder the vpright line.

Thē I consider euerie number, comparing it to the number that is againste it, and by­cause I finde them to be euerie one like hys match, I know that I haue wel done.

This crosse I perceiue doeth serue for those thrée denominations, pounds, shil­linges, pennies. But what if I had ob, q̄, q, and c?

You thinke you be at Oxforde stil, you bring forth so faste your q and c. These lines, as I haue saide, doe serue for thrée de­nominations, such as they be: as here they do serue for poundes, shillings, and pennies: but if yée haue no poundes in your summe, then may they serue for shillings, pennies, and halfe pennies: yea for q, q and c, if you haue no greter denomination, so that you remem­ber that the vpright line serueth for the grea­test denomination, and the highest thwarte line, for the nexte, and the lowest for the least,

And so if you haue foure denominations [Page] you must make your crosse with

so many lines. And if that your summe be of more denominati­ons, make so many lines in your crosse. And thus will I make an ende of Ad­dition.

Examples of Addition. [...]

The Proofes. [...]

An other Example. [...]

SVBTRACTION.

THen haue I learned the two first kindes of Arithmetike: nowe as I remember, doeth fo­lowe Subtraction, whose name me thin­keth doth sounde con­trarie to Addition.

So is it in déede: for as Addition encreaseth one grosse summe by bringing manye into one, so contrarie waies, Sub­traction diminisheth a grosse summe by with­drawing of other from it, so that Subtracti­on or Rebating is nothing else, but an art to withdrawe and abate one summe from an o­ther, that the Remainer may appeare.

What do you call the Remai­ner?

That you maye perceyue by the name.

So me thinketh: but yet it is good to aske the trouth of all such things, leaste in trusting to myne owne coniecture, I bée de­ceyued.

So is it the surest waye. And as I sée cause, I wyll still declare thyngs vnto you so plainelie, that you shall not neede to doubte. Howbeit, if I doe ouer­passe it sometimes (as the manner of men is to forget the small knowledge of them to whome they speake) then do you putte me in remembraunce your selfe, and that way is surest.

And as for this worde that you laste asked me, take you this description:Remainer. The Remai­ner is a summe lefte after due Subtraction made, which declareth the excesse or differēce of the two other numbers: as if I woulde abate or subtract 14 out of 18, there should re­maine 4, which is called the remayner, and is the difference betwéen those two numbers 14 & 18.

I perceiue then what Subtracti­on is: Nowe resteth to knowe the order to worke it,

That shall you doe by this mea­nes. Firste you muste consider,☜ that if you should go about to rebate, you must haue two sundrie summes proposed, the firste which is your grosse summe or summe totall: (and it must be set highest) and then the rebatement [Page] or summe to be withdrawen, which must be set vnder the firste (whether it bée in one par­cel or in many) and that in suche sort, that the firste figures be one iuste ouer an other and so the seconde and thirde, and all other folo­wing, as you did in Addition: then shal you drawe vnder them a line, and so are your summes duelie set to beginne youre wor­king.

Then beginne you at the righte hande (as you did in Addition) and withdrawe the ne­ther number out of the higher, and if there remaine anye thing, write that righte vnder them beneth the line: and if ther remaine no­thing (by reason that y^e 2 figures were equal) then write vnder them a ciphar of noughte. And so doe you with all the other figures, e­uermore abating the lower out of the higher, and write vnder them the Remainer still, til you come to the ende. And so will there appeare vnder the line what remayneth of youre grosse summe, after you haue dedu­cted the other summe from it, as in this ex­ample.

I receiued of your father 48 s, of whiche I haue layde out for you 36 s: nowe woulde I knowe what doeth remaine? and therefore I [Page] set my numbers thus in order: First I write the greatest summe, and vnder him [...] the lesser, so that the figures at the right side be euen one vnder ano­ther, and so the other, thus.

Then do I rebate 6 out of 8, and [...] there resteth two, which I write vnder them right beneath the line, thus.

Then I go to the second figures, [...] and do rebate 3, out of 4, where there remaineth 1, which I write vnder them righte, and then the whole summe and operation ap­peareth thus.

Whereby it appeareth, that if I withdraw 36, out of 48, there remaineth 2.

Nowe will I proue in a greater summe: And I wil Subtract 2367924 out of 3468946. Those summes I set in order thus. [...]

Then doe I beginne at the righte side, and deducte 4 out of 6, and there resteth 2, whiche I write vn­der them. Then goe I to the seconde figures, and withdrawe 2 out of 4, and there remaine [Page] two, whiche I set vnder thē also: then I take 9 out of 9, and there resteth 0, which I write vnder them: for you say, that if the figures be equall, so that nothing remain, I must write this ciphar 0 vnder them.

It was well remembred, nowe go foorth.

Then I come to the fourth place and draw 7 out of 8, and there remayneth 1, which I write vnder them also. Then in the fifte place I take 6 from 6, and there resteth nought, for it I write vnder them a ciphar, 0: Then in the sixt place 3 rebated from 4, there remayneth 1, which I write vnder them: and likewise in the vij. & last place, 2 taken from 3, there is lefte 1, whiche I [...] write vnder them: so haue I done my whole working, and my summes appeare thus. Whereby I sée, that if I rebate 2367924, out of 3468946, there re­mayneth 1101022.

Thys is well done. And that you maye be sure to perceiue fullye the Art of Subtractiō. let me see how can you subtract 52984732 out of 8250003456.

Firste I sette downe the greatest [Page] summe, and after that I write vnder if the lesser number, beginning [...] at the righte syde: and then my figures will stand thus.

Then take I 2 from 6, and the reste is 4 whiche I write vnder them: then doe I withdrawe 3 from 5, and there remayne 2, which I write vnder them. Then take I 7 out of 4, but that I cannot, what shal I now doe?

Marke well what I shall tell you now,Note. how you shall doe in this case and in all other like. If any figure of the nether summe be greater than the figure of the summe that is ouer him, so that it cannot be taken out of the figure ouer him, then muste you put 10 to the ouer figure, and then consi­der how muche it is, and out of that whole summe withdrawe the nether figure, and write the rest vnder them. Can you remem­ber this?

Yes, that I trust I shall. Now then in mine example where I shoulde haue taken 7 out of 4 and coulde not, I put 10 to that 4, which maketh 14, from it I take a­waye, 7, and there resteth 7 also, whiche I [Page] write vnder them.

So haue you done well, but nowe muste you marke another thing also: that whensoeuer you doe so put 10 to any fi­gure of the ouer number, you must adde one stil to the figure or place that followeth next in the nether line, as in this example there followeth 4, to which you must put 1, and make him 5, & then go on as I haue taught you.

Then shall I say: 4 and 1 (which I must put to him for the 10 that I added to 4 before) make [...], which I should take out of 3, but that cannot be, therefore must I put to it also 10, & then it will be 13, from whiche I take 5, and there resteth 8 to be written vnder them: and because of that 10 added to the 3, I must ad 1 to 8 that followeth in the nether line, & that maketh 9, which I should take out of 0, and cannot, therefore I put thereto 10, and that maketh 10: from 10 I take 9, and there remayneth 1, whiche I write vnder them.

Then doe I adde likewise to the nexte figure beneath, which is 9, and that maketh 10, that 10 should I take out of the figure a­boue, but I cannot, for it is 0, therefore I [Page] put 10 to it, and so take I 10 out of 10, & there resteth 0 to be written vnder them. Then come I to the next figure which is 2, and to him doe I ad 1, which maketh 3, that 3 I can not take out of nought, therefore of y^t nought I make 10, and thence do I take 3, so remai­neth there 7 to be written vnder them. Like­wise do I put 1 to 5 that followeth, and then is it 6, that would I take out of 5, and can­not, therefore I adde 10 to that 5, and make it 15, from which I rebate 6, and there re­maineth 9, which I write vnder them. Now haue I spent all the nether figures, & what shall I doe more?

You should haue added to the next figure following (if there had bene any) because you added 10 to the last figure before of the ouer line: but seeing there is no figure following, you must adde that to the place following, and then deducte that from the number aboue.

Then shall I saye because I borrowed 10 to the ouer 5, I muste put in the next place beneath, that is vnder 2: then must I subtracte that [...] from [...], and there re­steth 1, to be written vnder that, in the ninth place. Nowe I haue no more to subtracte, [Page] for there is neuer any figure remayning be­neath, nether yet any vnite to be added, be­cause I borrowed not 10 to the figure laste before,☞ and yet is there 8 remayning in the ouer line, which (I thinke by reason) shoulde be set at the ende of the figures in the lowest row whiche is vnder the line, for because there was nothing taken from it.

That is wel considered, and rea­son teacheth so in dáede.

But syr I beséeche you, shall I alwayes when any number so remayneth a­lone (as this [...] did) write him vnder the lyne straight against his owne place?

Yea, what else? Whether they bée one or many: and this wel remembred, you haue sufficiently learned Subtractiō. How be it, because of certaine thinges that might deceiue you, if you did not take good héede to your working, I wil propose to you another example of many numbers to be subtracted, as thus,

I receyued of a friende of mine to kéepe [...]869 Crounes, of which at one time I deli­uered him againe 500, at one time 368, and at another time 440, and an other tyme 80, and an other time 64: nowe would I know [Page] how many doeth rest behinde. Therefore firste I set downe my [...] grosse sum, and a lyne vnder it: & vnderneath it I set all the parcels, thus: and vnder them a double line.

Then firste I begin at the first place, & ga­ther togither the sum of al those lynes (saue the ouermost) in their first figures, and so do I with all the figures of the second place, & so forth as I did in Ad­dition, saue that I leaue out the highest row of numbers (as the line warneth me) and that summe so gathered betwéen the double line, doe I subtract out of the highest row of numbers, and the remayner doe I sette vn­der y^e nethermost line: [...] as for example.

I set the summes as before: than doe I ga­ther the first figures to­gether, where I fynde but 4 & 8, that make 12, (for thrée Cyphers increaseth no summe [Page] in addition, as you learned before) of the 12 therefore doe I write the digite 2, betwéene the double line, and kéepe the article in my mind, til I come to the seconde place, where I finde 6, 8, 4, 6, that make 24, to them I put the article in my mind, and it is 25, of which I write 5 vnder the seconde place, and kéepe the digite 2 in my minde for the third place, where I finde 4, 3, 5, that make 12, to the which I adde the 2 in my minde, and it ma­keth 14, thereof I write the 4 vnder the thirde place: and because there remayneth no more figures to be added, I write the digit 1 in the fourth place, as you see in the example.

Then come I to subtracting of this sūme betwéene the lines, for by Addition it is e­quall to the fiue parcels ouer it. Therefore I procéede to subtracte it from the ouermoste summe, saying: 2 from 9, remayne 7, to bée written vnder thē beneath the lowest line. Then in the seconde place I take 5 from 6, and there resteth, to be written vnder thē. Then in the third place, 4 from 8, resteth 4. Last of all in the fourth place, 1 from 2, re­mayneth 1. And thus I sée that after those 5 summes are subtracted from 2869, the Re­mainer [Page] is 1417.

This I perceyue: but is there no shorter way and more spéedier?

Yes,An a­bridgemēt of the for­mer maner of Subtra­ction. when you are a while ex­ercised in it: for you may as fast as you can gather the numbers togither, withdrawe them out of the highest sūme if so be it, that all the parcels which you doe gather, do not excéede nine, but and if they excéede nyne, then must you subtract onelye the digit that is in it, and reserue the article till the nexte place, where you shall adde it with the other figures and so subtract the whole out of the figure aboue them: but and if in this place the summe of the parcels do excéede 9, then (as I sayd before) subtract the digit only, and reserue the article to the nexte place, and so still goe forth, till you haue ended your working.

As for example: In the last summe pro­posed, I gather firste in the firste place 4 and 8, that maketh 12, of whiche I deducte the digitte 2 out of 9, and write vnder the remayner, which is 7, and the article 1 I kéepe in my minde. Then in the second place I gather the parcels 6, 8, 4, 6, which amount to 24, to that I adde the article 1, which I [Page] haue in my minde, and then is it 25, then do I take 5 (that is the digitte in this number) frō 6, that is in the seconde place of the high­est summe, and there remayneth but 1 to bée written vnder them, and nowe doe I kéepe the article 2 in my minde still. Then in the thirde place, 4, 3, 5, maketh 12, and the arti­cle 2 in my minde maketh 14: then take I 4 (which is the digitte) from 8, that is ouer them, and there resteth 4, whiche I write vnder them. Then haue I the article 1 yet in my minde, which I shoulde adde to the par­cels next following, but séeing there is no number following, I take that digit alone, and deduct him out of the next sūme aboue, which is 2, and then is the remayner 1, which I write in the fourth place vnder 2. Loe, now haue you a shorter way.

I like both wayes well, and I perceiue both well, yet as in the one the working séemeth somewhat long, so in the other it leaneth very much (me séemeth) to remembrance, and therefore maye cause er­rour quickly, except a man haue a quick and an exercised remembraunce.

What? would you then haue suche a way that should not be so long as the [Page] one, nor so short as the other?

Yea, if there were any suche.

Than doe thus: [...] still as you gather your parcels, when they excéed a digit, & ma­keth him 10 or more, take the article, and write him betwéene two lines (as in the first example) vnder the next place towarde the left hande: and then deducte the digitte from the figure that is ouer him, and wryte the remayner. And then when you gather the next parcels, you shall adde to them the figure that is vnder them, betwéen the two lines. And if it excéede 9, do as I said before, write the article vnder y^e next place betwéen y^e lynes, and subtracte the digitte from the Figure that is ouer those parcelles: and if that all the parcelles togither and the num­ber betwéene the lines do make but a digit, then deduct it wholy from the figure aboue: as in this example. I would subtracte oute of 40308964, [...] these thrée parcelles

Therefore I set thē firste in order due: & [Page] then I gather the parcels of the firste place, which are 8, 2, 1, that is 11: of whiche I take away the article, and sette him vnder the se­cond place betweene the lines: and the digit 1 that remaineth, I deduct out of 4, and there resteth 3 to be written vnder the first place beneath the lowest line. Then come I to the second place, and gather the parcels of it, 6, 4, 2, and the 1 betweene the lines, which make 13, of whiche I take the article, and set him vnder the third place betwéen the lines, and the digit 3 I take from 6, and there remai­neth 3, which I write vnder the second place beneth the lowest line. Than in the thirde place I find 4, 3, 4. which with the 1 between the lines, doe make 12, therefore I write the article agayne vnder the fourth place, and the digit 2 I take from 9, and there remay­neth 7 which I write vnder them beneath the lowest line.

And thē come I to the fourth place, where I gather 1, 2, 3 & h [...] 1 betwéene the lines, that maketh 7, which because it is but a digitte I plucke from 8, and the Remayner is 1. and must be written vnder them in the fourthe place. After that come I to the fifte place, where are onely three cyphars, which make [Page] nothing, thē should I take that, that is to say nothing, from the figure ouer them, whiche is also a cyphre, therefore I must saye thus: if I take nought from nought, there remay­neth nought: so must I write a cyphre vnder them. Thē in y^e sixt place I find but 1, which I take out of 3 ouer him, and the Remainer is 2, that must be written beneth the lowest line in the sixt place. So I go to the seuenth, where I find only cyphres, and in the grosse sum ouer them a cyphre also, therefore must I write their remainer (whiche is nothing) with a cyphre also. Then in the eighte and last place, I gather 1, 1, 2, that make 4, which if I take out of that 4 that is ouer thē, there will nothing remaine. And that must be no­ted with a cyphre beneath the lowest line, as I haue often sayde, and so haue I ended my worke, and the figures [...] stand thus.

Syr, I re­member you taughte me that cyphers shoulde not come in the last place, for because they serue onelye to encrease the valewe of other Figures whiche followe [Page] them, and serue not for those figures that goe before them: and nowe in your exam­ple you haue set two ciphers in the two laste places.

I commende you for your re­membraunce. And truth it is, I should not haue set them here, but only because that I would make you plainly to perceiue the art of Subtraction. Therfore séeing that you do now perceiue it, whensoeuer you shoulde write downe a cyphar, looke whether anye other figures be yet behinde. And if not than let go the cyphar also, for it néedeth notto write him in any latter places, where no other figure doth followe, excepte it be (as I did) to teache the vse of Subtraction the playner.

Therefore my fi­gures [...] must stande thus when I haue ended my worke.

So I woulde thinke by y^t you taughte me before. And now I beléeue I could subtract any summes.

So may you if you haue mar­ked [Page] what I haue taught you. But because this thing (as all other) must be learned surely by often practise, I will propounde here two examples to you, wherein if you often exercise your selfe, you shalbe ripe and perfecte to subtract any other summe lightly, for in them is contained all the obseruaun­ces of whole number. And because you shal perceaue somewhat both how to doe it, and also whether it be well done when you haue prooued to doe it, therefore haue I written vnder them, both the Remayners: And to one of them also adioyned his proofe.

[...]

Sir, I thank you. But I thinke I might the better doe it, if you did shewe me the working of it.

Yea, but you must proue your selfe to doe some thinges that you were ne­uer taught, or else you shall not be able to doe any more than you were taughte: And that were rather to learne by rote (as they call it) then by reason. And agayne there is nothing in this example or anye other of whole number, but I haue taughte you the rules of them alreadie.

Then I trust by practise to at­taine the vse of it. And is this all that I shall learne of Subtraction?

Yea, sauing that (as you haue séene in Addition) there are numbers of di­uers denominations, in whiche the wor­king is not muche vnlike, yet without some instructions be giuen of it, it might séeme to a learner more difficulte, then in déede it is. Therefore I will brieflye shewe you the vse of it onely, by one example or two.

A certayne man owed to me 14 lb, 12 s, 8 d, of which he paide me at one tyme 4 lb, 6 s, 8 d: at an other time 3 lb, and at an other 2 lb, 3 s, 4 d, and last of all, 6 s, 8 d. Now would I know what remaineth vn­payde [Page] yet, therefore I set my [...] summes thus.

Syr, I pray you why doe you write 2 lb. for the common spéeche vseth rather to say 40 s.

We must here vse the denomination that is greatest in any summe, so that we maye not write according as we vse to speake, saying: 16 d. 18 d: or likewayes, 7 grotes, 8 grotes: 24 s. 40 s 48 s. and suche other, but we muste write euerye denomination that is in any summe by it self, namely shil­linges and poundes. So must we write for these summes now named, 1 s, 4 d: 1 s, 6 d: 2 s. 4 d: 2 s, 8 d: 1 lb, 4 .s: 2 lb, 8 s: and so foorth of other like.

So that we maye not write in Arithmetike pennies, when the summe a­mounteth to shillings, nor shillinges when tho sum maketh poundes. Nowe if it please you, ende your example.

When my summes are so set as I shewed, then must I begin with the smal­lest denomination, saying: 8, 4, 8, are 20. which summe because it is pence, & 12 pence [Page] doe make 1 s, I must take [...] from that 20 (which com­meth of the parcels) 12. & for them write betwéene y^e lines vnder y^e shillings, than the 8 d, that remay­neth I take out of y^e high­est sum, which is 8 also, & then remayneth nought: wherfore vnder the pence I write nothing. Then come I to the shil­linges, and gather the parcels 6, 3, 6. whiche with the 1 betwéene the lines, make 16, that must I take out of the summe that is ouer it. But séeing that summe is but 12, I cannot take 16 out of 12, I must borrow one of the 14 lb, and put to the 12, and that maketh 32, for 1 lb is worth 20 s, then take I 16 out of 32 and there resteth 16 to be written vnder the shillinges. Then come I to the poundes, whose parcels are 2, 3, 4. that is in all 9, and one more must I adde therto, because of the 1 that I borrowed before vnto the 12 s, and than is there 10, whiche I must take oute of 14, so doth there remain 4 to be written vn­der the poundes: so doeth my remainer ap­peare to be 4 lb, 16 s.

This doe I perceiue verie well, and if there be none other thing to be learned in Subtraction, then maye I come to Multi­plication, for that you reckened to bée in or­der next.

We haue done in déede wyth the arte of Subtraction, as touching the wor­king. But yet before we go to multiplication,A proofe of Subtra­ction in numbers of one deno­mination. I wil instruct you how to examine your work whether it be well done or no, and that is by casting away 9 as oftē as you can finde it, as you did in Addition, sauing that you muste here examine the highest number alone, and note the residue of it at a lines end, as you did in Addition.

And when you haue done with the highest number, then examine all the other togither, casting thence 9 as often as you can: and if the last remayner be like the other, then haue you done wel.

But if you haue diuerse denominations in your summe,A proofe in Subtra­ction amōg diuerse de­nomina­tions. yet for them all shall you make but one seuerall line, as you did in Addition, remembring to begin the examination at the greatest denomination, and to double the re­mayner of pounds, and triple the remainer of shillings, as you did also in Addition.

[Page]As for a proofe, I will exa­min [...] this work where in the highest line I finde of poūds 14. from thence I bate 9, & there resteth 5, which I doe double, because they are pounds, and thē are they 10 thereto I ad the 12 & it ma­keth 22 frō whiche I take 9 twise, & there resteth 4, whiche because they are shillinges, I triple, and then are they 12, thereto I adde the 8, & then are they 20 thēce take I twise 9, and yet resteth 2, whiche I write at the one ende of a line thus, 2—

Then I examine all the other parcelles & the remayner together, euery denominatiō by it selfe. And first of poundes I finde 4, 3, 2, 4 that is 13. from which I take 9, & there resteth 4. that doe I double, and it maketh 8, to it doe I put the shillinges, 6, 3, 6.16. that is 31 (for the one betwéen the lines must not be reckoned nor none in that space) and that maketh in all 39. Where hence I take 9, foure times, and there remayneth 3. y^t doe I take thrée times, and it is 9, wherfore I cast it away: then doe I take the pennies 8, 4, 8. that maketh 20, from which I take 9 twise, [Page] and there resteth 2. whiche I write at the o­ther end of the proofe lyne. And bicause I sée that those two numbers are equall, I saye that I haue well wrought.

And if you will you may make for euerye denomination a line, as you learned in Ad­dition: but then must you beginne your ex­amination at the smallest denomination, as you did in Addition, for their proofe is alto­gither like, sauing that in Addition you ex­amine the nethermost summe alone, and all the other togither: and in Subtraction yée must examine the highest number alone, & al the other togither. And if you marke it wel, it is euen all one, for that summe that in Ad­dition is lowest, in Subtraction is highest:Grosse or total sume▪ and that summe is called the Grosse or To­tall summe.

Therefore if you marke what I sayde in Addition, you may easilye perceiue what is to be done for the proofe of Subtractiō. And to the intent that you may perceaue it y^e bet­ter, I will shew you an other proofe of Sub­traction, and that shall be by Addition, thus. Draw vnder y^e lowest nūber (which is your remayner) a line: then ad that number,An other proofe of Subtractiō. and all the other that you did subtract before, to­gether, [Page] and write that that amounteth, vnder the lowest line: and if the summe that com­meth thereof, be equall to the highest of the subtraction, than was the [...] subtraction well wrought, or else not. As for exam­ple: in the laste summes, which stoode thus.

First I adde 8, 4, 8, that maketh 20. whereof I take 12 awaye, bycause they make one shilling, and write for them vn­der the shillings: and the 8 that is lefte, I write beneath the lowest line, then adde I the shillings 6.3.6.1.16. that make 32: from whiche I take 20, and for it I write 1 vnder the poundes, and the 12 that remayneth, I write vnder the shillings. Then come I to the poundes, adding them togither, which are 4.3 2.1.4. that maketh 14: then doe I write 14 vnder the lb, and so haue I ended the Ad­dition. And I sée that the lowest line of num­ber and the highest be like, wherfore I know that I haue wel done. For my figures ap­peare as you may plainlie sée in the page fo­lowing.

[Page]And thus nowe haue I [...] taughte you the arte of Subtraction, and the means to proue whether it be well wroughte or not.

And this last proofe of Subtraction is most ap­test and beste allowed of any other proofe: whe­ther it be of lb s d. or any other grosse some whatsoeuer.

Nowe and you remember, I omitted in teaching the proofe of Addition one way whi: che I said was by subtraction.

Trueth it is, and then was it de­ferred, because that I had not thē learned the feate of Subtraction, whereby I should haue proued it, but now I thanke you, I haue wel learned the arte of Subtraction, & the proues of it, both by 9. and by addition. And nowe I would be glad to know, howe I may proue Addition by Subtraction.

Then marke you this.The profe of Additi­on by Sub­traction. Whē you haue ended your addition: take the numbers all that you did adde, to the highest summe, and deduct or subtract them from the grosse [Page] summe that doth resulte, and if the remay­ner be like to the highest number, then haue you done well, else not.

As for example. I take one of the summes that I did adde before, which [...] was this that followeth here.

Then doe I come to the middle number (because here in this example are onelye thrée numbers) and subtracte that from the nether nūber, beginning at the right hande, and first I saye, 0 out of 0, there remaineth 0: that write I vnder an other line. Then againe, 0 in the second place from 0, remai­neth 0 vnder it I write 0 also. Next that in the third place, 4 out of 2 wil not be, therfore I ad to that 2, 10, and make it 12 from that I take 4 and there resteth 8. Then saye I farther: 9 in the fourth place, and 1 (whiche I must adde for the 10, borrowed before) make 10, that must I take from 6: and be­cause I canne not, I adde to the 6.10, and then is it 16: from then I take 10, and there resteth 6. to be written vnder them. Againe in the fift place where I finde nothing writ­ten, I must sette 1 for the 10 last borrowed, and that 1 doe I take from the 1 vnder him, [Page] and so remaineth noughte, wherefore I write downe a cypher 0. Now haue I done with the subtraction: and yet in the grosse summe remayneth 1, whiche I must sette right in the same place, in the remayner, and so the remayner appeareth to bee lyke vnto the highest summe of [...] the Addition, as here ap­peareth. Wherefore I say that the Additiō was wel wroughte. And note, that if you had subtracted the vppermost from the product or totall sūme, then the residue thereof woulde be equall to that middlemost number. But if the parcelles which you added, be more then two: (as thrée foure, fiue, sixe, or more) than from your grosse or totall summe subtracte first one of the percels: and note that newe residue. Out of that new residue, subtract an other of your percels, (whiche you will) and Note that second new residue. And if you haue no mo percels added, but three, than is that second new residue equall & alike to the thirde parcel, whiche you haue not (as yet) subtracted, if you haue wrought wel: both in your first Addition, and now in youre sub­tracting. [Page] And so in this wise, (if you haue foure, fiue, or more parcels) maye you pro­céede to make your selfe sure of your totall summe, first, by Addition of the saide par­cels, produced and gathered. And thus may you doe in any other summe of one denomi­nation or many: Sauing that I wil tel you by the way, the laste manner of proofe that was shewed you in Addition, is the beste, and the aptest proofe for the Rule of Addi­tion.

Sir, I thanke you moste hear­tily,

Therefore nowe will I make an ende of Subtraction, and will instructe you in Multiplication.☞

MVLTIPLICATION.

MVltiplication is such an operation, that by two summes produ­ceth the third: whiche third summe so many times shall containe the firste, as there are vnities in the se­cond. [Page] And it serueth in the steade of manye Additions. As for example. When I would know how many are 30 times 48: if I should ad 48, thirtie times, it would be a lōg work. Therfore was this worke of Multiplication deuised, which shall doe that at once, that Ad­dition should do at many times.

I perceiue the commoditie of it partlie, but I shall not sée the full profit of it, till I know the whole vse of it. Therefore sir I beséech you, teache me the working of it.

So I iudge it beste, but because that greate summes can not be multiplyed, but by the multiplication of digits, therefore I thinke it best to shewe you first the way of multiplying them: As when I saye, 8 times 8, or 8 times 9. &c. And as for the small digits vnder 5, it were but follie to teach any rule, seing they are so easie, that euerie childe can doe it. But for the multiplication of the grea­ter digits, thus shal you do.

Firste set your digittes one ouer the other righte, then from the vppermost downward, and from the neathermost vpwarde, drawe straighte lines, so that they make a crosse, commonlie called Saincte Andrewes crosse, [Page] as you sée here. Then looke how many eche of them lacketh of 10, and write that a­gaynst eche of them,The diffe­rence. at the end of the lynes, and that is called the Diffe­rences, [...] as if I would know how many are 7 times 8, I muste write those digittes thus.

Then doe I looke how much 8 doth differ from 10, and I [...] finde it to be 2, that 2 doe I write at the right hande of 8, at the end of the line, thus.

After that, I take the diffe­rence of 7 like­wise [...] from 10, that is 3, and I write that at the righte side of 7, as you see in this example.

Then doe I drawe a lyne vnder them, as in Addition, thus.

Laste of all I multiplie the two differen­ces, saying: 2 times [...] make 6, that muste I euer set vnder the differences, beneth y^e line: then must I take the one of the differences (whiche I will, for all is like) from the o­ther digit (not from his own) as the lines of [Page] the crosse warne me, and that [...] that is left, must I write vn­der the digits. As in this ex­ample. If I take 2 from 7, or 3 from 8, there remaineth 5: that 5 must I write vnder the digits: and then there appeareth the multi­plication of 7 times 8, to be 56. And so like­wyse of anye other digits, if they be aboue 5, for if they be vnder 5, then will their diffe­rences be greater than themselfe, so that they cannot be taken out of them. And a­gayne, such little summes euerye childe can multiply, as to say: 2 times 3, or 4 times 5, and such like.

Truth it is. And séeing me sée­meth that I vnderstand the multiplying of the greater digits, I will proue by an exam­ple how I can doe it. I would knowe howe many are 9 times 6.

It is all one in value to saye 9 times 6, or 6 times 9: but yet the order is best to put the lesse summe firste, saying: 6 times 9, and so of al other summes.

Then would I know, howe [...] many are 6 times 9: therefore I set y^e digits thus, & make the crosse thus.

[Page]Then doe I set their differences [...] at the right side: the difference of 9 which is againste it, and the dif­ference of 6, whiche is 4 againste it also, as in this example.

And vnder them I drawe a line. Then doe I multiplie the [...] digites togither, saying: one tyme 4 maketh 4, that 4 doe I write vn­der the differences thus.

Then take I one of the diffe­rences from the other digite, as one from 6, or else 4 from 9, and eache wayes there resteth 5, which [...] I do write vnder the digits thus. And so appeareth the multiplica­tion of 6 times 9, to bée 54. Thus I see the feate of this man­ner of multiplication of digittes.

Now mighte you goe straight to the multiplication of greater numbers, saue that both for your ease and suertie in wor­king, I wil draw you here a table, whereby shall appeare the multiplication of all digits, and this is it that followeth in the next page.

[Page]

In which figure, when you would knowe the producte in anye multiplication of digits, séeke your first or last digit in the greater fi­gures, and from it go righte forthe towarde the right hande, til you come vnder the num­ber of your second digite, which is in the high­est rowe: and then the number that is in the méeting of the rowes of little squares (whiche some directlie from both your propounded di­gits) is the multiplication that amounteth of them. As if I woulde knowe by this table the multiplication of 7 times 9, séeke first 7 in the greater figures, and then go right foorthe to­ward the right hand, till you come vnder 9 of [Page] the highest rowe, in which place where y [...] so come vnder the other digit (as here for ex­ample you come vnder 9) is alwaies contai­ned the of come or product, which you séeke and that place we tearme to be in the com­mon angle, in respect of the two numbers [...] taken on the outsides, as here in that com­mon angle, where y^e rewes of little squares (directly procéeding from 7 and 9) do méete▪ you haue 63, which 63 is the summe of the multiplication of 9 by 7.

This is very good and ready [...]. And so may I find the multiplication of any digits. But now how shall I doe in greater summes?

When you would multiplie a­ny summe by an other, you shal marke that it is the méetest order to set the greatest nū ­ber highest, which is the place of the nūber y^t must be multiplied: and likewyse the lesser number vnder it, for that is the place of the Multiplier or multiplicatour, that is to say, the nūber by which multiplicatiō is made: and is in Englishe alwayes put before this worde, Tymes: in suche speaking when I say, 20 times 70. And the number that fol­loweth this word, Times, is that whiche [Page] must be multiplied.

Therefore when I would multiplie one number by another, I must write the grea­test highest, and the lesser vnder it, as in Ad­dition. And vnder them must I draw a line. As for example: If I would multiplie 264 by 29, I must set them thus. [...]

Then must I multiplie e­uery figure of the higher row by euery figure of the nether rowe: and that that amounteth, I must set vnder the line, as thus. First I doe multiplie 4 by [...] 9, saying: 9 times 4 (or 4 times 9, whiche is all one) and that maketh 36, as the table before of digittes doeth declare, of that 36 I muste write the 6 that is the digitte, vnder the 9, and the 3 in the next place towarde the lefte hande.

Then come I to the seconde figure of the higher rowe, and saye: 9 times 6 make 54, of which I write the 4 vnder the 3, and the 5 vnder the next place (as the rea­son [...] willeth me) thus.

After that come I to the next figure, which is 2, and do mul­tiply it by 9, & that maketh 18: [Page] whereof I write 8 vnder the [...] thirde place, and the article 1 in the fourth place, thus.

And so haue I ended the first figure of y^e multiplier. Wher­fore I giue it now a fine dash with my pen.

Then beginne I with the [...] nexte figure, and multiply it into all the higher figures, as thus,

☞First, 2 times 4 make 8, that doe I write vnder the second place: for euermore the digite or first figure of multiplication that amoun­teth of the first figure of the higher number, must be set vnder the multiplier of it, and the other in their order, toward the left hand.

I vnderstande you thus: that the digit of the summe amounting of the multi­plication of the first figure of the higher row, by the firste figure of the lower row or multi­plyer, must be set vnder the first place: & that that amounteth of the same first figure by the seconde multiplier, muste be set vnder the se­conde place, & so of the other, if there be more multiplyers.

So meane I indéede: and if [Page] there amount but a digit, then must it be set vnder the multiplyer.

And now to go forth: I multiplie by the same 2, the second figure of the higher rowe, which is 6, saying: 2 times 6, [...] make 12: whereof I write the digit 2 vnder the thirde place, & the artice 1, I write vnder the fourth place.

Then do I multiplye the last figure of the higher sum, by that same 2, saying: 2 times 2 is 4: whiche I write vnder the [...] fourth place. And so haue I ended the whole multipli­cation: wherefore I also giue the 2 a dashe with my pen, thus: and so I do euer assoone as I haue dispatched any digit by whiche I multiplie. And the summs stand thus.

Than must I draw a line [...] vnder all those summes that amount of the multiplicati­cation, and must ad all them into one summe, as in this example you may sée.

[Page]Where in the first place I find but 6, and therefore write I it vnder the line. Then in the seconde place 843, make 15, whereof I write 5, and kéepe one in my minde, and so forth, as you learned in Addition. And so ap­peareth the whole summe to be 7656, which amounteth of the multiplication of 264 by 29.

If there be no more to be ob­serued in it, then can I do it, I suppose, as by this example I shall proue. I would [...] multiplie 1365, by 236, wherefore I set them thus.

Then doe I multiplie 5 by 6, saying: 6 times 5 make 30: of whiche I write the ciphre in the first place, [...] and the article 3 in the seconde place.

Then do I by the same 6. mul­tiplie [...] the seconde fygure of the higher summe, whiche is 6, say­inge: 6 tymes 6 make 36: of which I write the 6 vnder the se­conde place, and the 3 vnder the thirde place.

[Page]Then doe I multiplie the [...] thirde figure whiche is 3, by the same 6, and that maketh 18: of that I sette the 8 vn­der the thirde place, and the 1 in the fourth place.

Then come I to the last [...] Figure of the higher sum, and multiplie it by 6, say­ing: 6 tymes 1 make 6: that doe I write vnder the fourth place. And so haue I ended the firste multiplier, and dash him slightly with my pen.

Then begin I with the se­cond [...] multiplier, and say first 3 times 5, that maketh 15, of which I sette the 5 vnder the second place, because that the multiplier is there, and the 1 I set vnder the thirde place.

Then come I to the seconde [...] figure, that is 6, and multiplie it by 3, whiche maketh 18, of which I sette the 8 vnder the thirde place, and the article 1 in the fourth place.

[Page]Than come I to the third [...] figure, which is 3, and multi­ply it by 3, saying: 3 times 3, make 9, whiche because it is but one digit, I set vnder the fourth place.

And then comming to the laste figure 1, I multiplye it by 3, and it maketh 3. which I set in the [...] fift place, and then haue I [...]nded two of the multiply­ers, and the summes stande thus. And then I giue 3 his dashe.

[...] Then come I to the thirde multiplier, and multiplye it into euery figure of the higher summe, and firste I saye: 2 times 5 makes 10, of whiche I set the cipher vnder the mul­tiplyer in the third place, and the article 1 in the fourthe place.

And so multiplying the seconde figure 6, by that same 2, there amounteth 12: where­of [Page] I write the digitte 2, vnder [...] the fourth place, and the arti­cle 1, vnder in the fift place.

Nowe doe I multiplye by the thirde figure of the high­er summe, which is 3, and that maketh 6: which I set vnder the fifte place, as ap­peareth in the example following.

[...] Than come I to the last place, and multiplye that 1 by 2, and there a­mounteth 2, which I sette in the sixte place, and then doth the sum stande thus.

And so haue I [...] ended the whole multipli­cation.

But now (as you taughte mee) to knowe what this whole summe is, I must ad all those parcelles together, and then vnder the line will appeare as you maye see the grosse or totall summe, that is, 322140.

That is well done.

Then me thinketh I would call it well done, when I knew whether I had well done or no.

It may be tryed by 9, as additi­on was, but the surest proofe is by Diuision, and therefore I will reserue that till you haue learned the art of Deuision.

And before we passe from Multiplicati­on I wil yet shew you another way of mul­tiplication, which is counted of some men, and is in déede, both more readyer, and more certayne, whiche differeth nothing from this that I haue taughte you, saue that it doeth vnderstande alwayes the ar­ticles, and ioyne them to the nexte summe, and therefore I will declare it onelye by an example.

If I woulde multiplie 1542, by 365, I must set them as I sayde before, and then do I multiply 2 by 5, and it maketh 10, [...] of which I write the article vnder the firste place, and kéepe the digit 1 in my mind.

Then saye I forth: 5 times 4 doe make 20, and the 1 in my minde, are 21, [Page] thereof I write the 1 vnder the [...] second place, and kéepe the 2 in my minde.

Then come I to the third fi­gure [...] 5 saying: 5 times 5 make 25, & the 2 in my minde make 27, whereof I write the 7 vn­der the thirde place, and kéepe the article 2 in my mynd.

Then comming to the last figure, [...] I saye: 5 times 1 make 5, and 2 in my mind make 7: that doe I write vnder the fourth place.

And then haue I ended my firste multiplier, and therefore I dashe it. Then doe I likewise with the second mul­tiplier, saying: 6 times 2 make 12, [...] therof I write the digit 2 vnder the seconde place, and keepe the article [...] in my minde.

Then say I forth: 6 times 4 ma­keth [...] 24 and 1 in my minde make 25. so I set that 5 vnder the thirde place, and kéepe the 2 in my minde.

Then multiplie I forth, saying: 6 times 5, maketh 30, and 2 in minde make 32, whereof I write [Page] the 2 vnder the fourth place and, [...] kéepe the 5 in my minde.

Then do I multiplie the last fi­gure [...] 1 by 6, and it maketh 6, to y^e I adde the 3 in my minde, and it maketh 9, whiche I write in the fift place.

And so haue I ended ij. figures of the multiplier.

Than with the thirde & [...] last multiplier, do I like­wayes, and say first: 3 times 2 make 6: which I write in the thirde place vnder the multiplyer.

[...] Than by that 3 doe I multiplye likewaies the seconde figure 4, and it maketh 12, whereof I write the digitte 2 vnder the fourth place, and the article 1 I kéepe in minde.

Than come I to the thirde fi­gure [...] 5 saying: 3 times 5 maketh 15, and the 1 in my minde make 16, thereof I write the 6 vnder the fift place, and kéepe the ar­ticle 1 in my minde.

[Page]Then come I to the last figure, which is 1 and multiply it by 3, & it ma­keth [...] 3, thereto I adde the 1 in my mynde, and it maketh 4, which I write in the 6 place. And then haue I ended y^e mul­tiplicatiō, and the figures stand in order thus.

Which parcels if I adde into one summe, it will be 562830, which is the grosse or total sum of al that multiplication

Well, this maner of multipli­catiō I perceaue: but what other sorts haue you?

There is one way that is wrought

by a checker Table, made thus.

Looke howe manye places your sūme hath that you woulde mul­tiplie so many squares must you make in your table, from the righte side to the lefte:Another vvay of multiplica­tion. and so manye from the higher parte to the lower, as there be places in your multipli­er. Then set downe your greatest summe first on the toppe of the table, euery figure in due order, in a square alone: I mean in those [Page] squares that be opē and vncrossed. And like­wise in those like squares at the right hand, set downe your multiplicatour or multipli­er, the last figure in the highest place, and so downward, that the first figure maye bée in the lowest place.

Sir if it please you, me thin­keth then I vnderstand you best, when you doe not stand long in telling the rule before examples: But propose, some example, and then in declaring it, bring in the rules withall.

In déede, that way is easiest for a yong learner, therefore will I euen so doe. Take this example: now I would multi­plie 2 [...]36 by 2,

First I consider that my

greatest number hath foure figures or places, & therfore I make so manye roumes betwéene lines, thus. Than I sée that of my multipliers there

are two, wherefore I drawe so manye lines a crosse the o­ther, that there maye be two roumes betwéene them thus.

But you must not forget to [Page] let the endes of the lines run out, as if ap­peareth in this Patron, for in those open squares must your two first numbers, and al the total summe be set.

Thē draw a crosse bar through euerye close square, so that it maye reache down to the lo­west ouerthwart lyne, as in this form. And than is your checker forme prepared.

Then sette downe your firste or greatest summe on the top, [...] & your multiplyer on y^e right side in y^e open squares thus.

Then begin to multiplie the first figure of the higher sum, by the highest of the multi­plier, saying: 2 times 6 make 12, that 12 must you write in the square that is against the 2 [...] and the 6, but in suche manner that the digit be sette in the nether corner of the square, & the article in the higher corner: as you may sée in this example.

And so of euery other multiplicatiō, what euer amounteth you muste write in the [Page] common square, which is against both those figures, by which you doe multiplie. And if that summe doe make but one digitte, than must it be set in y^t lower corner of y^e square, but if it make an article, than write the ar­ticle in the higher corner, and let the cipher go (if you will) euermore, for here it serueth for nothing, seing the lines doe distincte the places: but if the summe amounting of such multiplication doe make a mixte number, then write the article in the higher corner, and the digitte in the lower corner, as I did by that 2.

Then when you haue multiplied and en­ded the first figure, come to the nexte, & mul­tiplie it in like maner, as in saying: 2 times 3, is 6: that 6, because [...] it is but a digitt, you shal set in the nether corner of the square, next vnder 3, thus.

Then go forth, saying: 2 times 0 is 0: set [...] that vnder the barre (if you list) in the third square.

Then forth and say: [...] times 2 make 4, that set in the last square vnder the [Page] barre, so haue you ended the first multiply­er: Dashe him.

Come nowe to the seconde multiplier, and saye: 3 times 6, make 18, of whiche summe, the article 1 must [...] be set aboue the barre, in the square that is nexte to the 3 (as you sée here) and the 8 vnder the barre.

Then say 3 times thrée make 9, set it in y^e next suqare beneth the bar. Then 3 times 0 is 0. write it in the next square, or let it go, for all is one.

I perceiue it well [...] for here the lines distinct the places, wherefore ciphers do only serue, and therefore here they neede not to be.

Then saye farther: 3 tymes 2, make 6: write that in the [...] last square, then will the whole figure stand thus.

Now could I (me séemeth) do like againe. But how shal I do now to gather the summe?

Marke firste the order of the places in this figure, and so shall you perceaue the reason of gathering them into a summe.

[Page]The sloape barres do part the places; so y^t the first place is the lowest corner in al such figures (of the nethermost square nexte the right hand: & all the halfe squares betwéene that barre and the next, standeth for the se­cond place, and so the roome betwéene that & the next bar, is the third place: & so forth. Now if you perceaue this, then must you ad all the figures of one place together, as if you had an Addition of diuers summes.

If I vnderstand you right, then must I take here in this example 8 to be in the first place: 9, 1 and 2, in the second: 0, 6, 1, in the third: 6, 0, in the fourth: 4 in the fift: and the sixt place hath no figure.

You say well, and the reason is bi­cause the multiplication seruing to y^e square, made but a digit.

Then it is all one, [...] as if they stoode thus.

Euen so it is: and now adde this sum, and there wil appeare the totall of the multiplication to be 46828.

A proofe vvithout squares.And if you will sée the agréement of this maner of multiplication, and the other that you learned before, then multiply those two [Page] summes (that is 2036, by 23) [...] after the first maner without squares.

You meane to set them thus in order.

And then multiplie 3 into 6 [...] make 18: 3 times 3 make 9, 3 times 0 is 0 then 3 times 2 make 6: whiche must be set thus.

Then do I likewise with the second mul­tiplier, saying: 2 times 6 make 12, 2 times 3 make 6 2 times 0 is 0, & 2 times 2 make 4, which when I adde to the other then will the whole multiplication stād [...] thus.

So that you may sée in euery place the same figures, as they were in the multipli­cation by squares, though they differ in heighte and lownesse of places, but being added to­gither, they make one summe.

And thus now ye haue learned thrée sorts of multiplicatiō, which liketh you best, that may you vse.

Yet are there other forms, but sith they no­thing [Page] differ from these thrée in effect, but only in setting of the nūbers, I will ouerpasse them till a more méeter place and time. And now wil I instruct you in Diuision, so that you thinke your self sufficiently to perceaue what I haue taught you.

Yes sir I thanke you, but I doe not perceaue how to examine my worke, to trie whether I haue well done or no.

That is commonly vsed by y^e proofe of 9, as you learned before in Addition and Subtraction, saue that it hath this wayes diuers from them.

Firste you must make a crosse after

this maner.

Then must you examine your sum that shoulde be multiplyed, and looke what remaineth after casting awaye of 9, that sette you at the one side of the crosse: then examine the multiplier, and whatsoe­uer remayneth in it, after casting away 9 as often as you can, write that at the other side of the crosse: then must you multiplie those two numbers togither, and looke what amounteth thereof, if it be vnder 9, write it at y^e higher part of y^e crosse: but if it be aboue 9, then take thence 9 as often as you canne, [Page] and write the reste at the heade of the crosse. As in the laste example of multiplication, the number to be multiplied is 2036, wherein is once 9, and 2 remaineth, which I [...] write at one side of a crosse thus.

Then doe I examine the mul­tiplier, [...] which is 23, wherin there is no 9 but 5 in al, that 5 therfore I set at the other side of the crosse, thus.

Then doe I multiplie by 2 and it maketh 10, from whiche I withdraw 9 and there re­steth 1 that 1 do I set at the head of the crosse, then doe I examine the grosse summe, amounting of the multiplication, whiche is 46828. wherein I finde 9 three times and 1 remayning, that .1. I set at the foote of the crosse and then I see it to agree [...] with the other 1 at the toppe of the crosse, and so know I that I haue done well: for if they two did differ, then were my worke vaine, and the multipli­cation false.

This is the common proofe, but the moste certaine proofe is by Diuision, of which I wil anone instruct you.

Sir, what is the chiefe vse of Mul­tiplication?

The vse of it is greater then you can yet vnderstand: howbeit, these plaine cō ­modities it hath, that if you would resolue a­nye great and whole valure into many small and lesse portions: as if you woulde chaunge poundes into shillings, pence or anye other greater or smaller parcels, by multiplication, yée shal do it spéedely and easily. Also if you shoulde néede to adde one summe to it selfe, or to anye other oftentimes, you shall doe it by Multiplication muche more spéedelie, readily, easily and surelye, then by often and sundrye Additions. Take you these commodities grossely shewed for an answere at this time, and hereafter I wil more abundantlye make you to perceiue the vse of it.

DIVISION.

WEll sir, then in Diui­sion I praye you to instructe mée. But me thinketh by the name of it, that it shoulde be al one with Multiplication: for I call that Diuision, when any thing is parted into diuerse & ma­ny partes.

You take it as it is taken com­monlye, howbeit, if you marke wel, you shall perceiue that it is quite contrary to Multipli­cation, and doth not parte one thing or fewe things into many, but contrary ways it brin­geth many parcels in to fewe, but yet so, that these fewe taken togither, are equal in valure to the other many: for by Diuision pence are tourned into shillings, and shillings into poundes: as for example of 1 [...]0 shillings, it [Page] maketh 6 pounds, so are 120 tourned into 6, which is a smaller number: but then if you consider the denominatours, you shal sée that they are such, that one of the latter is equal to 20 of the first, and so in value the summes are one, though in number they doe farre differ, and the latter summe is the lesser, and so is it alwaies in diuision how be it, yet in the wor­king, the summe is parted by an other, & ther­of doth it take the name.

I thinke I shal better vnderstand the reason of the name, when I know the vse of the worke, therefore now would I gladlye learne that.

Diuision vvhat it is.Diuision is a distributing of a greater summe by the vnities of a lesser Or Diuision is an Arithmetical producing of a thyrde number, in respect of two propounded numbers: which thirde number shal so oftē containe an vnit, as the greater of the two propounded numbers doth contain the lesser. So that, euen as Multiplication did séeme to serue in stéede of manye Additions, so Diui­sion maye seeme to be in place of many Sub­tractions: Bicause that third number briefly expresseth, how many times the lesser of your two propounded nūbers may be Subtracted, [Page] from y^e greater: As in practise wil more plain­ly appeare. Therefore (as you may perceyue) vnto Diuision are required three nūbers: the first, which should be diuided, and that muste (generally) be the greater: and the seconde, by which the other must be diuided, and that is (generallie) the lesser, & is called y^e Diuisor. And the thirde which aunswereth to the que­stion, How many times: and therefore is cal­led the Quotient.

The first must be firste written and the se­conde so set vnder it that the last figure of the lower number be right vnder the laste of the higher,A generall rule for placing the figures. contrarie ways to the worke of the o­ther kindes of Arithmetike: for in them the two first figures were set euer méete one vn­der the other, but in Diuision the last figures must be set méet, except it chance so,An excep­tion. y^t the last figure of the Diuisor be greater then y^e last of the higher number, for then you shal set the last of the Diuisor, vnder the last (saue one) of the higher number, as for example.

If you shoulde diuide 365 (whithe are the summe of the dayes of a yeare) [...] by 2 [...] which are the dayes of a common moneth, then shoulde you set them thus.

[Page]But if you would diuide those [...] 365 dayes, by 52. whiche is the number of wéekes in one yeare, then should you set them thus.

Likewayes if I would di­uide [...] the same 365 by 4, whiche is the summe of the quarters of a yeare, then muste I set them thus.

Syr, this doe I vnderstande, but howe nowe shoulde I do to diuide the one by the other?

You must beginne with the laste figure next the left hande,☞ and see howe many times the last figure of the diuisor may be ta­ken out of the last figure of the ouer number, and that shall you note within a crooked line toward your right hand. As for example.

I woulde diuide 365 by 28, [...] then set I those two summes thus.

And I looke howe manye times I may find 2 (which is the last figure of the diuisor) in 3, (which is the last of the num­ber to be diuided) and considering that I can take 2 out of 3 but once, I make a crooked line at the right hande of the numbers, & with [Page] in it I set 1, and that is called the Quotiente number, as I tolde you.Quotient number. Then bycause that when 2 is taken out of 3, there [...] remaineth 1, I must write that 1 ouer 3, and deface or cancell the 3 and the 2, then will the fi­gures stand thus.

Then must I goe to the nexte figure of the diuisor, and take it likewayes so many times out of the figures that be ouer it, and looke what doth remaine, that I must write ouer them, and cancel them, as in this example.

Therefore now I do take once 8 out of 16, and there remayneth 8, whiche I must set o­uer the 6, and cancel or crosse [...] out the 15. and the 8 of the diuisor: And then will the fi­gures stande thus.

And so haue I once wrought.

So I perceiue that you take the nether figure not only out of the other that is right ouer him, but out of that with the other also that remayneth before, and are written toward the left hand.

So must you doe: for you must so take the diuisor out of the ouer number, that there remain not ouer it so great a summe as [Page] it selfe is, for then were your worke in vaine.

But yet againe here must you marke, that when you seeke how many times the laste fi­gure of the diuisor may be found in the num­ber ouer him,Note. that you looke also whether you may as often finde al the figures folowing in those that are aboue them, (considering al the remainders if there be any) if not, take youre Quotient lesse by one and then proue again, and so stil, til you find a moete Quotient: And by that moete quotiēt must you alwaies mul­tiplye your diuisor, and the product set vnder your diuisor, so y^t his first figure stand vnder the firste figure of your diuisor, and the second vnder the second, and so foorth: and then sub­tract that product from the number to be di­uided, that standeth directlie ouer it, as you haue séene me do.

☞When you haue thus wrought once, than must you beginne againe and write your di­uisour a newe, nearer towarde [...] the righte hande by one place, as in this example, you shall sette 2 vnder 8, and 8 vnder 5, thus

Then (as before) séeke how many times you may take your diuisor out of [Page] the number ouer him now.

That may I do here 4 times.

Truth it is that you may finde 2, foure times in 8: but then marke whether you can finde the figure following so manye times in the other that is ouer him: Can you finde 8 foure times in 5?

No, neither yet once.

Therefore take [...] out of [...], once lesse.

That is 3 times.

Well, then 3 times 2 make 6:Marke hovv to consider this kind of remainer. if I take 6 out of 8, there remayneth 2: which 2 wyth the 5 following, make 25, in whiche summe I finde 8 in times also, and therefore I take 3 as a true quotient, and write [...] it within the crooked line of the quotient, before the one thus.

Then saye I: 3 times 2 [...] make 6, then 6 out of 8 resteth 2, therefore I cancel the 2 and the 8 and write ouer it the 2 y^t doth remaine, thus.

Then do I take 8 as many times out of 2 [...], saying: 3 times 8 make 24, and if I take 24 [Page] out of 25, there remaineth 1: [...] so then I cancell 25 and 8, and ouer the 5 I set 1, thus. Or you mought (after you founde 3 to be a fit quotiēt) straight way haue multi­plied the whole diuisor 28, by that 3, at once: which giueth 84, whiche being set vnder 28, and duelie subtracted from 85, [...] of the number diuided, gi­ueth 1, y^e remainer of y^e whole diuision: as before you hadde. Worke which waye you list: here you sée also the forme.

And now haue I done with diuiding, for I can finde my diuisor 28 no more in the ouer summe.

No, except you would part the 1 that remayneth, into 28 partes.

That is wel saide, and so must we doe in such cases, whē there remaineth any thing: but I wil let that passe nowe, and wil make you perfect in diuision of whole numbers, and wil hereafter teach you particularly of brokē numbers, called Fractions.

Now if you do perceiue the order of diui­sion, then do you diuide this summe, 136280 [Page] by 452.

Firste I set downe the number that should be diuided, then doe I set the diuisour vnder it, so that the laste figure of it be righte vnder the last figure of the [...] ouer number. Then wil it be thus.

Can you take the last of your diuisour (which is 4) out of 1, which is the last of the ouer number?

I had forgotten, because the last of the diuisor cannot be taken out of the laste of the ouer number, in so much as it is the greater, therefore must I set the diui­sor [...] one place more foreward, toward the right hande thus.

And then muste I looke how often I maye finde the last figure of the diuisour (that is 4) in 13, which thing I maye do 3 times, ther­fore doe I say: 3 times 4 is 12, whiche I take out of 13, and there remaineth 1. Then do I make at the right hande of my summes a cro­ked line, and write before it my quotient 3: and I cancell 13 and 4, [...] and ouer the 3 I set y^e 1 that remaineth, and thē the figures stand thus.

[Page]Then do I multiplie the same quotient in­to euerie figure of the diuisor, and withdraw the summe that amounteth out of y^e numbers ouer them, as first I saye: [...] times [...] make 15, which I take from 46, and there resteth 1, I cancell therefore 16 and 5, [...] and write ouer the 6 that 1 that remaineth, thus.

Then doe I say lyke­waies, 3 times 2 makes 6, whiche I take out of 12 and there resteth 6, therefore I cancell the 12 [...] and the 2. and ouer the 2 I write 6 that remay­neth, thus.

Then should I set for­ward [...] the diuisor, into the next place toward y^e right hand, thus.

But you maye sée, that ouer the 4 is no figure, therfore must I set the diuisor yet fore­warder by an other place.

And marke, when soeuer it chanceth so, y^t you should set forwarde the diuisor, and that it can not stande there, bycause there is no nū ­ber ouer the laste place, or if there be any, it is [Page] lesser then the laste figure of the diuisor,☜ then must you remoue the diuisor yet once again: and because that his firste place of remouing serued not to subtracte him so much as once, therefore shal you write in the quotient a Ci­pher 0. And if you should by chaunce néede to do so oftētimes, for euerie time write a cipher in the quoitente. The reason of this, will I shew you hereafter.

Then must I [...] set my summes thus.

And bicause I remoued [...] the diuisor, so y^t I ouer­skipped one place, A must write a cipher in the quo­tient: & then must I séeke a new quotient, as in this exāple I must say, How many times 4 is there in 6? and sith it can be but once, therfore do I write 1 in the quotient, and then saye I: 1 time 4; ta­kē [...] out of 6, remaineth 2, I cancel the 6 and the 4, & write 2 ouer thē thus.

Then saye I againe, once 5 out of 28, remai­neth [Page] 23, I let the 2 stande as it did, and ouer y^e 8 I set 3, cancelling the 8 [...] and the 5 vnder it, thus.

You might as well haue saide, once 5 out of 8. & so remayneth 3 but now go foorth.

Then once 2 out of 0, can not be, what shal I now doe?

Borow of the next number that is be­hinde (for there is 230) and doe as you lear­ned in Subtraction in a like case.

Then must I borrowe 1 of the 3 comming behinde nexte, and make that 0 to be 10: and then take I 2 out of 10, and there resteth 8. And because I borrowed one of the 3. I must cancel the 3, and [...] write 2 ouer it: then doth the figure stand thus.

Nowe haue you done, and yet remai­neth 228, and your quoti­ent shew you, that if you diuide 136280 by 452 you shal finde your diuisour in your gre­ter number 301. that is CCC times & once and 228 remaining.

And in the other example, where I diuided [Page] 365 by [...], the quotient was 13, & 1 remay­ned: whereby I know that in a yeare (which containeth 365 daies) there are 13 monthes reckening 28 dayes (or 4 wéekes) iuste to a moneth, and 1 day more.

Why then do we cal a yeare but 12 monethes.

Of that at a more conuenient time wil I fullie instruct you: but nowe it is not con­uenient to entangle your minde with other things, than do directlie pertaine to your mat­ter. Therefore if you remember what you haue hearde, you haue learned a short mone [...] of diuision, which I would haue you often to practise, so that you maie be perfect in it, and hereafter I wil shew you certaine other pro­per points touching it,

Then I pray you, yet tel me, how I shal examine and trie my worke, whether I haue done wel or no, that thoughe no man be by me to tell me, yet I may perceiue it my selfe.

Some men (yea and commonlie) do trie it by the rule of 9, as in all the other kinds, saue that their order is this. First they cast away 9, as often as they can, out of the diuisor, and that remaineth, they set at [Page] one side of a crosse: As in our first example, the diuisor was 28, from which you [...] may take 9 thrée times and 1 re­maineth which they set by a crosse, thus.

Then do they likewise examine the quo­tient (which in our example is 13) and from thence they cast away 9 as oftē as they can, and the remainer they set at the other side of the crosse, and then multiplie they toge­ther those ij. remainers: and to it that a­mounteth they adde the remainer of the di­uision; if there were any, from that whole summe they withdraw 9 as often as they [...]an, and the rest they set at the head of the crosse: as in our example the quotient is 13, from which take 9, and there remaineth on­ly 4 and therefore must you set 4 [...] at the other side of the crosse, thus.

Then multiplie 4 by 1, and it yieldeth but 4 thereto adde the remainer of the diuision (which was 1) and it will bée 5, which summe doth not amounte to 9, and therefore must be set wholy at [...] the head of the crosse, as you sée here.

And this number on the [Page] heade of the crosse, is the first proofe, to whych if you finde an other like in the number that was diuided, then haue you done wel.

Therefore now shal you likewise examine the whole summe that was diuided, and take away 9 as often as you can, and that that re­maineth, set at the foote of the crosse: and if it be equal to that in the head of the crosse, then haue you wel done, else not.

As in our example the whole summe was 365, which maketh 1 [...], from [...] that take 9, & there resteth 5, which set at the foote of the crosse, thus.

And you shal sée that they agrée: therefore haue you wel done.

Nowe will I likewise examine our second example, where y^e diuisor was 452 whiche maketh 11: from thence [...] I take 9, and the 2 that remai­neth I set at the right side of a crosse, thus.

Then examine I the quoti­tient, which was 301, where I finde but only 4. that do I set at the other side [...] of the crosse thus.

Then do I multiplie 4 by 2, [Page] and it maketh 8, to that do I ad the remay­ner of the diuision (which was 228, and ma­keth 12) and they two make 20, wherein I finde twise 9, and 2 remaining, that [...] 2 must I sette at the heade of the crosse, thus.

Then doe I examine the whole number to be diuided, which was 136280, where I finde twise 9, and 2 re­maining, [...] which I set at the foote of the crosse, thus.

And because that it doeth agrée with the figure at the head of the crosse, I know that the diuision was well wrought.

This is the common proofe, howbe­it, the more certaine working is by the con­trary kinde, as to prooue Diuision by mul­tiplication, thus.

Multiplie the quotient by the diuisor, and if the summe that amounteth be equall to the sum that should be diuided, then haue you well diuided, else not.

Howbeit, this must you marke, y^t if there remained any thing after the diuision, that muste you adde to the sum that amounteth of the multiplication: as in our first exāple [Page] the quotient was 13, and the diuisor was 28: Now multiplie the one by the other, & the summe will bée 364: to that if you adde the one that remained after the diuision, than will it be 363, which was the summe that should be diuided, & therefore I knowe that I haue well done.

Now will I prooue the same in the second example, whose diuisor was 452, and the quotient 301: these doe I multiplye together, and there amounteth 136052, to which if I adde the 228 that remayned, then will it be 136280, which was the whole summe to be diuided, and therefore I per­ceiue that I haue well done.

This is the surest way to examine Diuision by Multiplication: and contrarie waies the surest proofe of Multiplication, is Diuision.

And therefore now wil I shew how you may proue Multiplication by Diuision.

When you haue ended multiplicatiō,A proofe of Multi­plication by Diuisi­on. and would know whether you haue well done or not, set the grosse sum that amounteth of the Multiplication ouermost, and diuide it by the multiplier: and if the quotient be the same number that shoulde be multiplied, [Page] then haue you well wrought, else not: as in that example where we multiplied 264, by 29, the grosse summe was 7656.

Nowe if you will knowe whether that multiplication be true, you shall diuide that 7656 by the multiplier, 29: and you shall perceiue that the quotient will bée 264, and that is a token that you haue well wrought.

By your patience I wil proue that: and first I set down the grosse summe & the multiplier, not after the rule of multi­plication, but after the rule of diuision, for now that nūber is become the diuisor, that was before the multiplier, I shall [...] set them therefore thus.

Then shall I séeke howe many times 2 in 7, that may be 3 times, and 1 remayneth: but then maye not 9 bée found so often in 16, therefore must I take a lesser quotient, that is to say, 2 [...] then say I twise 2 maketh 4, which I take out of 7, & there remaineth 3, then do I cancell 7 and 2, and ouer 7 I write 3 and in [...] the quotient I set 2, so the fi­gures stand thus.

Then say I forth, 2 times [Page] 9 make 18 which I bate out of 36, and there resteth 18 then cancell I 3 and ouer him set 1, and likewise I cancell 6 and 9, and ouer them I set 8, so that thus stand [...] the figures.

Then doe I sette forwarde the diuisor by one place, and séeke a newe quotient, that is to saye, how many times 2 are in 18, which I finde to be 9 times, but then can I not finde 9 so many times in 5, therefore I take a lesser quotient, as to say, 8: but yet is that to greate, for if I take 8 times 2 out of 18, there remaineth but 2 and I can not finde 8 tymes 9 in 25, therefore yet I take a lesser quotient, that is 7, which is also to great, for if I take 7 times 2 out of 18, there resteth 4, but nowe I cannot take▪ 7 tymes 9 out of 45, therefore yet I séeke a lesser quotient, as to saye, 6: then saye I▪ 6 times 2, make 2, that I [...] take out of 18, and there re­maineth 6, so I cancell the 18 & the 2 and write 6 ouer 8, thus

Then say I forth: 6 times 9 maketh 54, that take I out of 65 and there remaineth 11 [Page] [...] and the figures stand thus.

Then must I set forth the diuisor againe, and séeke a newe quotient, whiche will be 4: for though I maye finde 2 in 11 fiue tymes, and 1 remayne, [...] yet I cannot finde 9 so often in 16, therefore I set the figures thus.

And y^e 4 in the quotiēt I multiplie into the fi­gures of the diuisor, saying: 4 times 2 ma­keth 8, which I take out of [...] 11, & there resteth 3, there­fore I cancell the 11 & the 2 and set 3 ouer the firste place of 11, thus.

And then doe I say forth, 4 times 9 maketh 36, which I take frō 36, and there remaineth nothing, so that y^e quo­tient of this diuisiō, where 7656 is diuided by 29, is 264, which doth declare that if 264 be multiplied by 29, the sum will be 7656. And thus I perceiue now how both Multi­plication is proued by Diuision, & diuision also by Multiplication.

Nowe haue I ended the fiue cōmon kinds of Arithmetike: for as toucing Mediation, Duplation, triplation, and such other, they are no seueral kinds of Arithme­tike, but are contained vnder the other: for Mediation is contained vnder Diuision, & is nothing else but diuiding by 2: and so are duplation and Triplation contained vnder Multiplication: for Duplation is nothing else but multiplying by 2, and triplation is multiplying by 3, of which I wil only pro­pose an exāple,An exam­ple of Me­diation. for the rules you haue heard alreadie.

If you woulde mediate or diuide into [...], this sum, 4531010, you shal set 2 for the di­uisor, & work as you lear­ned [...] before, as thus.

Then I finde 2 in 4 two times, therefore my quotient must be 2: so I cancell 4 & 2, and remoue the diuisor [...] forward, thus.

Then againe I finde 2 in 5 twise, and 1 remaining, so I write 2 againe for my secōd num­ber [...] of the quotiente, and cancell 5 and 2, and o­uer 5 I set 1, thus.

[Page]Then remoue I the diuisor forward and séeke a new quotient, which is 6: then saye I, 6 times 2 make 12, take that out of 13, and there resteth 1, so I [...] cancell 2 and 13, and ouer 3 I sette 1 thus. Then remoue I the diuisor forward; and séeke a new quotient which is 5. then take I twise [...] out of it; and there resteth 1, so I cancell the 2 and the [...] last figure of 1, and let the firste stande thus.

Then remoue I the diuisor forward, and séeke a newe quo­tient, which is 5: then take I 2 fiue times out of 10, and there resteth nothing.

Then remooue [...] I againe the di­uisor forwarde, thus.

But bicause I can not finde the diuisor in the number ouer it, I muste sette a cipher in the quotiente, and remooue [Page] the diuisor to the [...] next place, as ap­peareth in the fi­gure before.

Then séeke I a new quotient which I finde to be 5, for so many times maye I haue 2 in 10. Then haue I fully ended this Mediation or Di­uision by 2, and the quotiēt is this, 2265505, which is the halfe of 4531010, as you maye trie by Duplation: for double that quotiēt,Duplation. or multiplie it by 2, and the same number will amount.

I will no longer tarry about these, séeing they are but members of the other kindes.Easie for­mes of multipli­cation. But here now wil I teache you certayn ea­sie formes both of Multiplication & of diui­sion, and first of multiplication.

If you would therfore multiplie any sum by 10, you shall néede to doe no more but ad a cypher before his first place, as for exāple: 36 multiplied by 10, make 360.

Likewise if you wil multiplie any summe by 100 put two ciphers at his beginning.

So if you would multiplie any sum by a thousand, ad thrée ciphers to the beginning of it.

This doe I well perceiue, and also the reason of it.

I will omit all reasons till our next méeting, when I shall tell you the reason of all other partes of Arithmetike also: and as to our matter now, looke (as I haue tolde you) that you both remember it, and also of­ten practise it.

But if you would multiplie any number by 5, marke first whether the number be e­uen or odde: and if it be euen take the halfe of it, and write a cipher at the beginning of it, as for example: I would multiplie 2564 by 5, I take the half of it, which is 1282 (as you may know by Mediation) and before it I set a cipher, thus, 12820, and this is fiue times 2564.

And thus may you do with any other euē summe, that you would multiplie by 5.

But if the summe be od, as for example 2563, thē must you take the lesser half of it, or (if you wil) take away 1 from the first fi­gure (as here take 1 from 3) & then take the halfe of the rest, and sette before it 5: as of 2563, the lesser half is 1281, for here I take but 1 for the half of 3: and if I put 5 before that lesser halfe, then haue I multiplied it 5 [Page] times, as thus, 12815.

What meane you by the lesser halfe.

There is no iust half of any od num­ber, therefore if we diuide an od number in­to 2 parts, as nigh equall as can be, yet will the one halfe excéede the other halfe by one, as for example. The two nearest halfs of 9, are 5 and 4: and likewise of 15, are 7, & 8, where you sée that y^e one part still is greater then y^e other by 1. Now it is easie to knowe which is the greater halfe, and which is the lesser halfe.

Then I perceaue you, and can doe likewise (I doubt not) with anye summe. For if it be not very easie to part into halfs, then will I doe it by Mediation easilye y­nough.

That is a sure way. And now haue you learned howe to multiplie easilie by 5, 10, 100, 1000: and of like maner maye you doe with any other of that sort.

But now if you will multiplie by 20, 30, 40, and so forth: or by 200, 300, and suche like, where there is one cipher in the firste place, or many orderly in y^e first places, you shal take away those ciphers, and multiplie [Page] the sum only by the other figure or figures, (if they be many) & then at the beginning of the sum that amounteth, shal you set so ma­ny cyphers as you tooke away.

Example of 2873, which I would multi­plie by 300. First I cast away the 2 ciphers from the multiplier, & I multiplie the sum by only 3 that is left, and it amounteth to 8619: before which I put the two ciphers that I toke awaye before, and then is it 861900. And that is the summe that amoū ­teth, when 2873 is multiplied by 300.

And if there were two or more figures beside the ciphers, I must only take away the cyphers, and multiplie by y^e other figures, as I learned before: as if I woulde multiplie 93648 by 25000, I should take away the thrée cyphers, and multiplie the same by 25, and then at the beginning of that totall summe, should I adde the thrée cyphers againe.

Euen so: but and if it chaunce the number that should be multiplied, or both the summes as wel the number that should be multiplyed as the multiplier, to haue ciphers in their first places, euermore caste away the Ciphers, and worke by the rest. [Page] But remember to restore as many ciphers to y^e amounting sūme, as you bated before, as in this example: 30200 shall be multi­plied by 206: I shall only take awaye the two ciphers from the greater number, and then multiplie 302 by 206, and afterwarde ad the two cyphers againe. But if I would multiplie the same 30200 by 2060, I shall not onely take awaye the two ciphers from the number that should be multiplied, but also I may take away the one cipher from the multiplier, and then must I adde 3 Ci­phers to the summe that amounteth: but take héed that you take away no cipher that commeth after any signifying figure, as in this last example, you may not take awaye that in the fourth place of the higher num­ber, neither that in the third place of y^e mul­tiplier: how be it, yet this you may doe: If one cipher or more come in the middest of your summes, you may multiplie by y^e other figures, and ouerskippe [...] them, but so, that you giue euerye figure his due place, as thus: I will multiplie 3026 by 2004, therfore I set them thus.

[Page]And thus doe I multiplie then: Firste 4 times 6 make 24: I sette the 4 vnder the first place, and kéepe the 2 still in my mind: Then saye I againe: 4 times 2 maketh 8, and the 2 that is in my minde, maketh 10, I sette downe the Cypher 0, and kéepe the article one in my minde. Then 3 tymes 4, is 12, and the 1 in my minde maketh 13, I sette downe the Figure 3, and kéeping the 1 still in my minde hauing no more places of y^e vpper number to mul­tiplie it withall: I put it downe nexte 2 in the fift place.

But now when I come to the next place being a Cipher 0: I let it go, because it multiplieth nothing: And likewise the se­cond cipher.

But then when I doe come to the 2, and multiplie it into the 6 of the ouer number, you must take héede (according as I taught you in multiplication) that the first number amounting of the multipli­cation, [...] be sette righte vn­der the multiplier, and the other orderly towarde the left hand, according as you may sée in this example. [Page] whiche being finished with the Addition thereof gathered togither, will stand as this example sheweth.

Which is indéede wrought [...] so muche sooner and shor­ter by ouerskipping of the two ciphers whiche else woulde haue hadde two workinges more as by the same example here also set down doth appeare.

Sir, I thanke you: for I sée greate ease in this waye of Multiplicati­on, and if you canne shewe mée suche like in Diuision, you shall greatlye further me.

Yes, I will teache you some easie ways in diuision also, and first this: If you would diuide any summe by 10, you shall onely with your penne make a square line,Easie forme [...] of Diuision. betwéen the first figure of your summe and the seconde, and than haue you done: for the whole number that followeth the line, standeth for the quotient and the figure that is before the line, is the remainer: as for example, 3648 diuided by 10, [...] will stand thus.

[Page]Where 364 is the quotient, and betoke­neth that so many times are 10 in 3648: and the 8 after the line, is the remainer, which cannot be diuided into 10, but by breaking it into fractions, wherwith I wil not meddle yet.

And so likewise if you would diuide any summe by 100, with your pen, you shal cut away the two first figures, & if ye would di­uide by 1000, you must cut away the 3 first figures, & so of any other diuisor, whose last figure is 1, & the other be ciphers, looke how many ciphers the diuisor hath, and so many figures at the beginning shal you cut away with the squire line, and they stand alwaies for the remainer because they are lesse than the diuisor, and cannot be diuided by it, and the other figures that be behinde y^e line, stād for the Quotient.

But now if your diuisor haue any other fi­gure in his last place than 1, and in al his o­ther places haue ciphers, looke howe manye ciphers there be, cut away so many of y^e first figures of the nūber that should be diuided, and diuide the rest that followeth the line, by that figure that is in the last place, as if it were the whole diuisor.

[Page]Example of 64284, whiche I would di­uide by 300, here must I cut away the two first figures, (for so many ciphers my diuisor hath) and must diuide the rest by 3 whiche is the figure in the last place of the diuisor. First therefore I part away the two firste [...] figures, & the summe standeth thus.

Then do I diuide 642, by 3, and the quotient is 214, for in 6 I finde twice 3, & in 4 once, and 1 remayning, which 1 with the 2 next before, doth make 1 [...], wherein I finde 3 foure times: and this is a readie way to turne shillings into pounds: for sith one pound doth contain 20 shillings, I must diuide the whole nūber of shillings by 20 therfore easilie to do it, I see that my diuisor hath one ciphre, and therefore I cut away one figure from the be­ginning of the whole summe of shillings, & then I do mediate or diuide by 2 the other fi­gures or summe that foloweth.

I wil put an example.

If you would diuide 64287 shillings by 20, that is to say: If I would turne so many shil­lings into poundes, I must cast away y^e first figure, that is 7. & diuide the rest, that is 6428 by 2 so shal the Quotient be 3214, whereby I know that 64287 shillings, make 3 [...]14 poūds, [Page] and 7 shillings remayning.

Nowe proue by Multiplication whe­ther you haue wel done or no.

The quotient is 3214. whiche I doe multiplie by the diuisor 2, and it doth amoūt to 6428.

Hereby may you perceiue not onlie that you haue wel done, but also how by diui­siō you may turn shillings easilie into poūds: And contrarie waies, by Multiplication you may turne pounds into shillings.

But here shal you sée amongst diuers men, diuerse formes of suche diuision, but if you marke what I haue tolde you, you shall per­ceiue easilie al their waies: for some men doe not cut away so manie of the first figures of the sum y^t they would diuide,An other manner of the abridg­ment. as there are cy­phers in the firste places of their diuisor, but they set al their ciphers orderly vnder the first places of the number that they would diuide, and then with the other figure (or figures if they be manye) they diuide the reste of their summe. Example. If they [...] would diuide 725931, by 3400, they set their sums thus.

And then doe they diuide orderlie till they [Page] come to the ciphers: for there they stay & ende their worke, as in this example: They seeke how often 3 may be founde in 7, which is 2 times, and one remayning, therefore they set 2 in the quotient, and can­cel [...] 3 and 7, & ouer 7 they set y^e 1 y^t remained, thus.

Then do they go forthe [...] saying: two times 4 ma­keth 8, whiche they take out of 12, and there re­maineth 4, thus.

Then renew they the diuisor forward, and séeke how often 3 maye be founde in 4, which is but once, and 1 remaineth, then set they 1 in the quotient, and can­cell [...] 3 and 4, & ouer them they set that 1, thus.

Then take they once 4 [...] out of 15, & there resteth 11. Or else more easilie: Take once 4 out of 5, & there resteth 1, so they cancel the 4 & 5, and set 1 ouer them, thus.

Then set they forth y^e diuisor againe, & seeke [Page] how manye times 3 are in 11, whiche they finde 3 times, and 2 remayning: so they set 3 in the quotient, and [...] cancell 11 and 3, and ouer them setteth 2, thus.

Then do they mul­tiplie [...] 4 by 3, whiche maketh 12, that with drawe they out of 29, and there resteth 17. of whiche the 7 must be set ouer the 9, and the 1 ouer the 2, thus.

And now are the two ciphers next ensuing, so that the diuisor can no more be set forward, and therefore is the diuision ended, and the remainer is 1731.

Nowe the quotient, which is 213, doth de­clare, that if you diuide 725931, by 3400▪ you shall finde it therein 213 times, and there remayneth 1731, so shall you finde it, if you worke as I taught you, by cutting away the two firste figures,☞ bycause of the two ciphers. But this must you marke (as you maye per­ceiue by this last example) that if there be left any other remainer in the summe that was [Page] behinde the squire line, that the remayner must be set to the latter ende of the firste re­mayner, whiche was cut [...] awaye wyth the squire line: as if you woulde diuide 725931 by 3400, after the forme that I taught you, then woulde your summes appeare thus.

So that 17, whiche remayneth after the line, must be sette to the 31 (that was cut a­way wyth the line) in higher places, as you sée here: where that 17 with the 31, do make 1731.

And here woulde I make an ende of Diui­sion, sauing that there commeth to my minde one late inuention of easie Diuision whiche I will brieflie set foorthe to you,An other inuention of easie Diuision. so that if you finde ease in it, you maye vse it. Bicause that the hardest point in Diuision, is the rea­die and easie finding of the quotient number: and againe, if that be truely knowen, all the rest is but light to be done: therfore this way­es shall you quickelye and truelye finde the quotient.

[Page]Firsts write the nine figures of [...] number: I meane 123456789, not a long as I haue set them now, but vp and downe as in this forme. And at the left side of them draw a long line, as you sée here: Then consider the diuisor, by whiche you intende to worke, and sette it on the lefte side of the long line, righte againste 1, and for a distin­ction drawe a line vnder it: then multiplye your diuisor orderlie by eache of those figures, beginning with 2, and so goe downewarde till you haue ended all. And looke what doth amount of the multiplicatiō of each figure into the diuisor, then write it a­gainst the figure wherby you did multiply.

By example I may perceiue it better.

Take this example [...] 263845 to be diuided by 64 thē must I set the 9 figures as I saide before, and the diuisour must I set against the 1, thus.

Thē must I multiply y^e diui­sor byech figure orderly: first by 2 and it maketh 128, which I must sette against 2 at the left [Page] hande.

Then multiplie 64 by 3, and it maketh 192, whiche is sette against 3. Then 4 times 64, make 256, y^e set I by 4 Then saie I, 5 times 64 make 320, that set I againste 5. Then 6 times 64, make 384, that sette I againste 6. Then 7 times 64 make 448, whiche I set a­gainst 7.

Further I say: 8 times 64, [...] make 512, which I set by 8. And last of all I saye: 9 times 64, make 576, whiche I sette againste 9. And then the will stand thus.

And so is the table ended, by which you may easilie find the Quotient, as you shal sée by example now.

Do you set down y^e numbers (as you lear­ned before) according to the order of diuision.

That is thus. [...]

Now looke what number standeth ouer the diuisor, reckning therto al them that be behinde it toward the left hand.

Then are there ouer the diuisour, 263.

That is iuste: nowe séeke in the table on the lefte side, whether you can finde 263.

It is not there.

Then take the number that is next to it, beneath it: I meane a lesser number than 263, but of all the lesser numbers that the table hath, take you that that goeth nigh­est to 263.

That is 256.

So is it: and marke this euer­more, when you can not finde iustlie in the table that summe that is ouer your diuisour, then note that that is nexte beneath it of any summe that is in the table,☞ and looke at the righte hande of the line what figure or digit that is against that summe, and take that di­git for your quotient, and then worke on, as you learned before: for now haue I tolde you the whole vse of this table.

Howbeit, yet that you maye be sure to vn­derstande it, I will sée you end thys example of Diuision by it.

Nowe therefore begin againe.

First I sette downe [...] the summes after the common maner, thus.

[Page]Then do I looke ouer the diuisor, and [...]nd there 263▪ Now to know howe many times 64 may be taken out of 263; I resorte to the table aforesaid; and seeke for the number 263; but it is not there, therefore as you [...]ad me, I take a lesser number, the nexte to it that I can finde in the table, and that is 256, which number hath againste it on the righte hande this digite 4 which I must take for the first figure of my quotient.

Then do I (as I learned before) multiply that quotient into eneme figure of the diuisor orderly, withdrawing the summe thereof, a­mounting out of the ouer sum: as here I say first: 4 times 6 make 24, so I take y^t out of 26, saying: 4 out of 6 remaineth 2, whych I write ouer y^e 6: then 2 out of 2 remaineth no­thing, then cancel I 2 and 6, [...] and also 6 in the diuisor, and the summe standeth thus.

Then doe I likewise say forth: 4 times 4 make 16. which I take out of 23, & there resteth 7 to be set ouer 3, & that 3 with the 2 behinde it and [...] the 4 vnder it, must be can­celled, as you sée here.

Then haue I done wyth [Page] the figure of the quotient.

Nowe sette forwarde your diui­sour, and séeke a newe [...] quotiente, as you sought this.

Then thus standeth the figures so that ouer the diuisour I sée 78, which I séeke in the table, and cānot finde it: therefore I take the next lesser, and that is 64 the diuisor it selfe.

So must you do when there is none o­ther.

Then againste [...] it I finde this digit 1, which I muste set in the quotient before 4, thus.

Then multiplie I 6 by 1, and it is but 6 stil.

Note.You néede not goe about to mul­tiplie when the Quotient is 1, for 1 doth nei­ther multiplie nor diuide, but in such case on­ly subtract the diuisor out of the number that is ouer it,

Then I take 4 out of 8, and there resteth 4, & 6 out of 7 there remayneth 1, so I [Page] cancel those numbres, & [...] write the remainers o­uer their places, thus.

Then set I forward the diuisor again, thus.

[...] Where I sée ouer the diuisour 144, whiche I séeke in the table, and finde it not: therefore I take the nūber in y^e ta­ble y^t is nexte thereto, beneathe it, whiche I finde to be 128, against which in the right side I finde 2, which I take for my quotient, and that doe I multiplie firste into 6, and thereof commeth 12, which I take out of 14, and then remaineth 2, that 2 I set [...] ouer 4, and cancell the other figures, 1, 4 and 6, thus.

Then say I for the: 2 [...] times 4 are 8, whiche I take out of 24, and there remaineth 16, of whiche I write the 6 ouer 4, & the 1 ouer 2 and cancell 2, 4 and 4, thus.

[Page]Now againe I set for­warde [...] the diuisor thus. And séeing ouer it 165 I séeke that in y^e table, but finde it not, therefore I take y^e next lesser, which is 128, against which I find [...]: that do I set into the quotiēt, and by it I [...] multiplie first 6, & ther­of cōmeth, 12, which I take out of 16 and ther resteth 4 thē cācel I 1, 6, and 6, & ouer 6 I set 4 thus.

Then do I multiply [...] 4 by 2, and it maketh 8, which I take out of 45, & there remaineth 37, as in this example.

And nowe haue I done.

Well, now I sée that you can worke by this kinde of diuision, as farre forth as I taught you.

Yea sir, I thanke you, and I find in it much ease and certainenes.

Yet one thing I doubt whether you per­ceiue: what if you did find in the table the nū ­ber that standeth ouer the diuisor, what wold you next do?

I thinke I shoulde take the digitte a­gainst it on the left hand for the quotient.

So is it: and as often as you seeke in the table and finde your number iust,Marke the diuersitie be­tvveene a true quotient and a iust quotient. the di­git against it is your true and iust quotiente. I cal that a true quotient also, if it be y^e right quotient that you shoulde take [...] though your diuisor multiplied by the same, do not cleare­ly subtract the number ouer it, but there doth somewhat remaine, as it chaunced in al the examples that you did worke by. But if it shoulde ch [...]nce (as it doth often) that your di­uisor multiplied by your quotient, do subtract clean the number ouer it, thē cal I that quo­tient not only a true quotient, but also a iuste quotient, because it doth iustlie consume the number ouer the diuisor: and that chaunceth euermore when the nūber ouer the diuisor is iustly found in the table.

This I shal remember.

But yet one easie pointe more I wyll tel you in this sort of diuision, therfore marke it wel.

[Page]When you haue founde in the table, other the same sum that is ouer the diuisor, other y^e next beneath, (for lack of the other) thē loke what digit standeth againste it, take that for your quotient. And bicause it is some paine to multiply the diuisor by the quotient, you shal not neede to doe it, but only take the number that you founde in the table, and subtract that from the ouer number: for if you do multiply the diuisor by the quotient, that will be the number that shal amounte, therefore is this way more easier.

So is it, and also more cetainer for such as I am, y^t might quicklie erre in multiplying, especiallie being smallye practised therein.

Thē proue in some brief exāple whether you can do it, and so wil we make an ende.

I would diuide [...] 38468 by 24, therefore first I set y^e table as here foloweth: Then set I the two sums of Diuision thus.

And ouer the diuisor I finde 38, which I seeke in the table & finde it not, therefore I take the nexte beneath it which the table hath, & y^t is 24, the diui­sor it self: against which is set [Page] 1, which I take for the quotient, which I set in his place. And now I néede not to multi­plie the diuisor by it, but only [...] to withdrawe the diuisor out of the 38, that is ouer it, & so remaineth 14, as thus.

Then set I forwarde the [...] diuisor, and finde ouer it 144, as appeareth [...]: then séeke I y^e number in the ta­ble and finde it, and against it is 6, therefore I set 6 before 1 for my quo­tient, and I take that 144 for the iuste mul­tiplication of the diuisor by that quotient, and therefore without any newe [...] multiplication I do subtract the 144, from the other 144, and there resteth nothing, as you may sée.

Therfore I set forward the diuisor: but se­ing it will not bée in the next place, (for then ouer 2 woulde be nothing) [...] I set it forwarde twice, as you sée here.

And for because that I could not set it in y^e nexte place following, therefore [Page] I set a cipher in the quotient, as you sée.

Then loke I ouer the diuisor, and find 68, which I can not finde in the table, therefore take I the next beneth it; which I finde in the table, and that is 48, and against it standeth 2, whiche I take for the quotient▪ And then without any multiplying of the quotient in­to the diuisor, I doe sub­tract [...] that 48 from 68, & there resteth 20, as here appeareth.

And so haue I ended the whole diuision.

In verie great summes to be di­uided by great diuisors; I thinke there is no better way thā this for any mā to vse, though he be neuer so expert. And that especiallie, if one great diuisor be oftē to be occupied about diuiding many and diuers great summes. As commonly happeneth in Astronomical wor­kings, and Geometricall, about the signes, both straight and reuersed: as if it be your for­tune and desire to wade to the profoundnes of Geometricall and Astronomicall calculati­ons demonstratiue, you will soone confesse. Whereof, an other time shall better serue to speake. Now can you sufficiently skil in these [Page] kinds of Arithmetike. And now for the far­ther vse of these two last, that is multipli­cation and diuision. I will briefly shew you the feate of Reduction, by the way.

REDVCTION.

REduction is, by which al summes of grosse denominati­on maye be turned into sums of more subtile denomina­tion. And contrarye waies, all summes of subtile denomi­nation, maye be brōught to sums of grosser denomination.

What cal you grosse denomination, and subtile denomination?

That I call a grosse denomination,G [...]osse de­nominatiō. whiche doeth containe vnder it many other subtiller or smaller: As a pound in respecte to shillinges, is a grosse denomination: for it is greater than shillings, and contayneth many of them. And shillings in comparison [Page] to poundes,Subtile de­nominatiō. or a subtile denomination, for because they are lesser then pounds, and many of them are contained in one of the o­ther: as so, likewyse of other thinges, whatsoeuer thing is compared to other, if it be greater and containeth many of them, it is a grosser denomination: but if it bée lesser, so that manye of them are in the o­ther, then are they called subtile denomina­tions: whereby you may perceaue, that one denomination may be called a grosse deno­mination, and also a subtile (that is to say, a great and small) in diuers comparisons. For shillinges compared to poundes are a Subtile or small denomination: but com­pared to pennies, they are a grosse or greate denomination.

Nowe I vnderstande the name, I pray you teache me the vse.

The vse is easily learned, if you re­mēber what you haue learned before. For if you will reduce any summe of a grosse de­nomination,To reduce grosse de­nominatiōs to subtile. into a summe of a smaller or subtiller denomination, you must consi­der howe many of that subtiler denomina­tion doe make one of the grosser denomina­tion, and by that number or numerator doe [Page] you multiplie the other summe: as if you woulde reduce 20 poundes into shillinges, you must consider that in a pound are inclu­ded 20 shillings, therfore multiplie the one 20 by the other 20, and there will amounte 400, whereby you may knowe, that in 20 pounde are contained 400 shillinges. Like­wise if you would reduce 30 shillinges into pennies, considering that in 1. shilling, are 12 pennies, you must multiplie 30 by 12, and it will be 360: whereby you find, that in 30 shillings, are contained 360 pennies. And thus may you reduce any grosse deno­mination into a more subtiller, by multi­plication, if you know how many of the les­ser doe make the greater: of which thing I will anone giue you a bréefe table for the most accustomed kinds of money, weights, measures, and tyme, and such like, whereby you maye knowe howe often eche subtile denomination is contained in the Grosser, when you shall néede it for the foresayde kinde of Reduction. And also the same shall serue you, if you would reduce any summe of a subtiler denomination,To reduce subtile de­nominatiō to grosse. into a summe of a grosser denomination: For in suche Reduction you must consider (as in the o­ther [Page] forme) howe manye of the smaller doe make the greater, and by that number must you diuide the other summe, and the quoti­ent will declare howe many of the greater denomination, are comprehended in that summe, as for example: If you woulde know how many shillinges are contayned in 3240 pence, consider that 12 pennies doe make 1 shilling: you must diuide that 3240 by 12, and your quotient wil be 270, wher­by you know that so many shillinges are in 3240 pennies. But and you would knowe farther, how many pounds are in those 270 shillinges, séing that euery pound contay­neth 20 shillinges, diuide that 270 by 30, and it will be 13, and 10 remaining, wher­by you may, know that in 3240 pennies, or 270 shillings, are 13 poundes and 10 shil­lings. For euermore the remainer muste be named by the name or denomination of the summe that was diuided, which in this place were shillinges.☞ And thus maye you doe with any other kindes of denomi­nations.

Wherefore to the intent that you maye haue a lighte knowledge in the common Coines, weights, measures, and such other, [Page] I haue prepared here a briefe table, whiche shall suffise to you at this time, til hereafter at more conueniente opportunitie I maye instruct you more exactlie in the same.

Note (gentle Reader) these valewes of Englishe comes, as they were when this Authour first publi­shed his Booke. But in our time (namelie An. 1582)▪ they are much diuerse. Therefore something to plea­sure thee in this purpose. I haue for thy benefit at the end of Reduction, set downe and annexed a Table, not onlie of our coines, but also of the most parte of Christendome, with their iust waighte and values, currant in this Realme of England, as by the same shall plainlie appeare.

• A Soueraine.
• Half a Soueraine.
• A Roiall.
• Halfe a Roiall.
• A quarter Roiall.
• An old Noble.
• Halfe an old Noble.
• An Angell.
• Halfe an Angell.
• A George Noble.
• Half a George Noble.
• A quarter Noble.
  Englishe Coines.
• A Croune.
• Halfe a Crowne.
• A Croune.
• A Grote.
• A harpe Grote.
• A pēnie of 2 pence.
• A dandie pratte.
• A pennie.
• An half pennie.
• A Farthing.

PROGRESSION.

ALthoughe vntill this day the most parte of writers haue defined Progression as a cō ­pendious kinde of Addition, yet truelye it is not so: for pro­gression (as the verie nature of the worde doth informe any man) is a going forwarde and proceeding in num­bers, and that regularlie and orderlie, whose place is aptlye chosen to be verie [...]eare, or ra­ther next after the exposition of the four prin­cipal partes of Arithmetike, for in it after a moste easie manner, are all the foure former partes exercised and practised: and not one­lye Addition, as customablie is done. Whiche custome hathe bene the cause, why it hathe so speciallye bene named a kinde of Addition▪ and defined to bée a quicke and briefe Addition of diuerse summes, procéeding by some certayne and reasonable order.

[Page]You shall also vnderstande, that there are infinite kinds of progressions, but for you (as yet) two are sufficiente to be exercised in: of which the one I cal Arithmeticall, and the o­ther Geometricall.

Arithmetical progression.Arithmetical progression is a rehearsing or placing downe of manye numbers, number after number, in such sort, that betwéene eue­rie two nexte numbers rehearsed or placed downe, the difference, diuersitie, or excesse, be equal and alike.

Syr, I thanke you for that you haue both opened vnto me what Progression is, truly, and also why it is here placed. But I pray you with an example make plaine your definition.

Examples can not want, séeing al rea­sonable creatures naturallie vse the order of one kind of Arithmetical progression (whiche therefore is also named Naturall) when so e­uer they distinctlie doe counte or number any multitude one by one, saying: 1.2.3.4.5.6. wherby the procéeding from number to num­ber, and euerie one surmounting and excée­ding his fellow next before by a like quantity (which here is 1) declareth the same to be A­rithmeticall [Page] progression. And for the more plainnesse, I set it down in this maner.

[...]

This is most euidēt. And I thinke that I am able to tel you now of any progression Arithmetical propounded, what is that com­mon excesse or difference wherby it procéedeth if this order be kept in it.

What say you of 3.6.9.12.15?

They excéede eche other by 3. And that may I set down in such euident order, as you did your example of Natural progression, in this wise.

[...]

And doe you not also nowe per­ceiue, that the whole table of Multiplication maye be made by the order of progression A­rithmetical? either if you wil begin at the first number of any of them on the left hande, and so procéede right ouerthwarte: or at any of the first numbers of the vpper rowe, and goe di­rectly [Page] downward?

I pray you let me consider the thing a little, and I wil answere you.

By this triall I perceiue it nowe verie well: for the common excesse or difference be­twéene any two next, is continually as much as the firste number of euerie rowe, either from the lefte hande ouerthwarte taken, or from any of the vppermost ouerthwart rows downward.

Nowe then if of any suche progression you woulde spéedelie knowe the totall summe, muche quicklier thā by common ad­ditions [Page] rules:To knovv the total sum of an Arith­meticall pro­gression. first tell howe many numbers there are (which numbers here we cal places or parcels) and if they be odde, write theyr summe downe by it selfe, as in this example, 2, 4, 6, 8, 10, 12, 14, where the numbers are 7, as you may sée, therefore set done 7 in a place alone: then adde togither the firste number and the laste, as in this example: adde 2 to 14, and that maketh 16, take halfe of it, and mul­tiplie by the 7, whiche you noted for the nū ­ber of the places, and the sum that amoun­teth, is the summe of all those figures added togither, as in this example: 8 multiplied by 7 make 56: and that is the summe of al the figures.

That will I worke by an other example. I would know how much this sum is, 5, 8, 11, 14, 17, 20, 23, 26, 29. I tel the pla­ces and they are 9, that I note. Then I putte the first number 5 and the last 29, togither, and they make 34. I take the halfe of it, that is, 17, and multiplie by 9, and it maketh 153. That you saye is the summe of all the numbers.

So shal you finde it if you trie it.

How shal I trie it?

By your common addition: for [Page] if you adde all the parcels togither, you shall see the same summe amounte, if you dydde worke well. And that manner of Additi­on trieth all kindes of summing anye Pro­gression.

Then can I summe a progression, if the numbers of the parcels be odde. But what if they be euen? as in this example, 1, 2, 3, 4, 5, 6, 7, 8?

When the number of the parcels is euen, then note that also as you did before, and likewise adde the first summe to the last, and by the halfe of the number of the places do you multiplie it: as in your example, the parcels are 8, that note I? then adding y^e firste summe to the last, there amounteth 9, that do I multiplie by the halfe of parcels, that is by 4, and it maketh 36, whiche is the summe of the 8 parcels.

☞But if you wil take one rule for these both, doe thus. Multiplie the halfe of the one by the other whole, and the summe wil amounte al one. For sometime it chanceth that the num­ber of the parcels be odde, so that their halfe can not be taken: and sometime it chaunceth the Addition of the firste number and the last, to bring forthe an odde number, so that the [Page] halfe of it can not be taken: but they will neuer be both odde.

Thē I perceiue this, if there be no more longing to it.

As accustomablie it hath bene taughte, thys hath bene the chiefe and onelie exercise in Progression vsed. But that you maye perceiue howe diuerse wayes and to howe great profitte so simple a thing (as this Arithmetical Progression is) maye be consi­dered and vsed, I will here propounde you sixe propositions, of which four of them were inuented by a friende of mine, and neuer be­fore this published: and the firste two, were neuer to my knowledge written of, but by thrée men.

This dothe greatlie encourage me to be attentife vnto your wordes, seeing I shall not onelye be instructed at your hands in the common knowen rules of this excellent art, but besids that, so aboundantlie in other new rules informed, as my verie entrance shall séeme to passe a great many mēs farther stu­die, and longer continuance. Therefore sir, I beséech you, let me know your sixe propositi­ons.

These they are.

• [Page]1 To know the last number without pro­ceeding by continual addition, til you come vnto it, [...]o that the common excesse, the first number and the number of the places bee knowne.
• 2 The first number of the Progression and the laste being knowen, with the com­mon excesse, to finde the number of the places.
• 3 The excesse being giuen, and the firste or laste, to know the quantitie of anye middle number, whose place is giuen from the first or last.
• 4 The totall summe being giuen, and the first and last, to finde out the number of the places.
• 5 The total sum of any Arithmetical pro­gression being giuen, and the first and last, to finde out the common excesse.
• 6 The totall summe being giuen, and the mutual excesse, with the number of the pla­ces, to giue the firste or laste number of the same progression.

Many moe considerations could I propoūd you in these Arithmeticall progressions, but these are sufficiēt to giue you occasiō to think, [Page] that rules of knowledge and artes are infi­nitely capable of enlargement.

Happie were I, if I did but well vnderstand that which is alreadie inuented and written. And yet in my simple fantasie. these thinges offer themselues (in manner) to be studied for about Progression, there­fore I pray you to procéede to the rules an­swering to these propositions.

I will orderlye for euerye of these sixe propositions giue you rules, and with euery one an example, vnlesse y^e plain­nesse and easinesse need no further exemple­fying.

For the Solution of the firste. Multiplie the excesse by a number lesse by 1 than the number of the places, and the offcome adde to the firste number, so shall you haue the laste number, whiche is soughte for.

As (for example) if there were seuen pla­ces in a progression Arithmeticall, whose continuall encrease, or mutual excesse were 5 and the first number were 5, and I would know what the last & seuenth number is, I multiplie 6, which is lesse then 7, (the nū ­ber of the places by 4, thereof commeth 24, [Page] whiche I adde to 5, that maketh 29: and that is the last number, whiche I desired to know. And this you maye straighte waye proue, by continuall procéeding from 5 till the seuenth place, encreasing euerye one by 4, as thus.

5 9 13 17 21 25 29.

Lo here, the last, being also the seuenth, is 29.

I perceiue already one good proper­tie in this rule, which in all workes is to be desired: y^t is, it wil ease one frō great labor, if a progression were propounded of 100 or 200 places, or moe, And also it is verye ea­sie to worke, and most necessarie for the to­tall summe finding, in a very long progres­sion.

The second rule is this. From the 2 last subtract the first, the remainer diuide by the common excesse, to the Quotient ad 1, and you haue the number of the places, which you wold know: As in this progres­sion.

6 11 16 21 26 31.

If I know only 6 and 31, and that they en­crease by 5, than according to the rule, from 31 I subtract 6, there remaineth 25, which [Page] 25 I diuide by 5 (the common excesse) the quotient commeth forth 5, to which I ad 1, that maketh 6: and so many are the places, as you sée.

This rule is so easie, that I were muche to blame, if I coulde not remember it.

The third proposition may alwaies 3 thus be soluted: Multiplye the excesse by a number lesse by 1. than the distaunce of the place is from the first or the last number gi­uen: the of come ad to the first, if the distance be reckned from the first, and the firste also known or subtract from the last: if the di­staunce be from the last counted, and y^e laste giuen also, and that whiche commeth forth, either in that addition to the first, or subtra­ction from the last, is the number sought As for example I propound you this progres­sion.

8 15 22 29 36 43 50 57.

And for the apt considering y^e maner of this questiō, I will note ouer euery place his di­stance from the first and vnder euery place his distance inclusiuely from the last, thus.

[Page]

Now, if that excesse whereby this Pro­gression standeth, be knowne to be 7, and the first number giuen, being 8, I woulde know what number standeth vnder 4, that is to say in the fourth place. I multiplie 7 by 3 (which is lesse by 1, than the number of the place propounded) that yéeldeth 21, to which I adde 8 (the first number) so com­meth 29: which I say to belōg to the fourth place, as ye sée in the example it also doeth: or if in the third place from the last, you would know what number in this example should stand, the last number being known to be 57, and the common excesse 7, than by 2 (which is lesse by 1 than the place propoū ­ded) I multiplie 7, that giueth 14: whiche I subtract from 57, so remaineth 43: which appertaineth to the thirde place inclusiuely reckned from the last, & so my example gi­ueth you.

I perceiue right good vse of this rule: for if I had forgotten what the firste number were, and remember still but the laste, the common excesse, and the number [Page] of the places, then mighte I come by the knowledge of my first number againe. And me thinketh, that it differeth not much from the first proposition sauing that which you make here a middle number, there was made the last: and also in this point it diffe­reth, that in it the last was only sought, and no consideration hadde in numbring the places from the last, as here I marke in your numbers noted vnder your progressi­on.

And thinke you not the mid­dle numbers of a Progression standing of a hundred or thrée hundred places or moe, may as much cumber a man to come to the knowledge of them by continuall encrea­sing from the first (by the common excesse) or abating from the last continually (the common excesse) as the very final numbers in a shorter Progression would doe?

Yes sir, that I thinke righte well, and therefore I am glad of this newe framed proposition, and the maner of the working of it.

The rule of the fourth is this.4 Adde the first and the last together, and by the ofcome diuide the totall summe. Dou­ble [Page] the quotient, and that will be the num­ber of the places.

Then if in a Progression, whose sum were 207, and the first number 12, and the last 57, if I adde 57 and 12 together, that maketh 69: and by it I diuide 207, the Quotient will be 3, which I double, and so I haue 6, and so many must be the number of the places, that this progression standeth on.

Whether it be so or no, howe will you trie?

Halfe 6, whiche is 3, being multi­plied by 69, must make 207, the totall sum, if 6 be the number of the places. For so the whole worke of your rule in summing any Arithmeticall progression did en­forme [...] me. I will than multiplie by 3, thus.

It commeth forth iustlye.

I must muche herein commende your promptnes, both in memorie and in well applying your rule: although in ma­nifest wordes if did contayne no suche mat­ter.

Sir, I praye you heare me frame one example more.

I am well pleased, so that ye be short, for you make me more longer here, then willinglie I would haue bene: but I can not perceaue how I could haue omitted any thing as yet, without your greate lacke thereof.

If I had receiued 85 pounds of cer­taine men but of how many▪ I haue forgot­ten, yet I remember that the first gaue mée 7 lb, and the last 27 lb, and euerye paymente after other did rise by a like sum. And the man for whom I receiued this money, con­ditioned with me, that of euery payment I should haue 12 pence for my labour: nowe vnlesse I can by arte find the trueth of this case, I am like to lose the moste parte of my reward.

I perceaue you can hansomly frame an example, which shoulde concerne your owne gayne: I praye you lette me sée howe you woulde doe Iustice in this poynte.

I adde the firste [...] and the laste together that maketh 34: by which I di­uide 85, thus.

Why how now? Sir, here [Page] is a remnant of 17, in which 34 cannot bée had, so that nowe I am in the briars for doubling of my quotient, and farewell then both my Iustice, and a good lumpe of my gaines.

Ye are neuer the farther from the matter, though it fall into a fraction. For you shall vnderstand that the fractiō which of any such worke procéedeth, is euer halfe of one such,☞ as the vnits of the quotient be­fore are. And that you may trie, if you double that which so remaineth, for then it will be equal to your diuisor, as if ye double 17 (the remnāt) it maketh 34, and your diuisor al­so was 34, this noteth the remainder to bée half of one.

Now I am glad of this hard exam­ple. For with it I haue a generall rule for the Fraction that maye hap in this worke. So that the quotient being two & a halfe, I double that, & it maketh 5, therefore shoulde my gaine be 5 shillinges. And to be sure (by your leaue) I will trie it for I [...] will multiplie halfe of 34, (whiche is the firste and laste number ioyned together) by 5 thus. It is moste true (I see) [Page] that I shoulde léese nothing by the former working.

The fift proposition hath this rule 5 appertaining vnto it: By the fourthe rule finde the number of the places, that béeing done, from the last subtract the firste, and the residue diuide by a number lesse by 1, than the number of the places, and the quotient will shewe the excesse whiche is sought for.

An example hereof shal be this: If ye had disbursed 685 lb. to a certaine number of men, you neither can tell howe manye they were or how muche the ones money excée­ded his next before, but you are sure that y^e excesse was equal betwéen euery two next: & also you remember that the firste had 19, and the last 118 poundes, how woulde you finde both the number of the men & the ex­cesse, continually obserued in the succession of their reiments.

Your rule doth plainlie bid, first to find the number of the places, [...] whiche I will doe according to she fourth rule. I adde 19 and 118 together, thus.

[Page]By this 137, I diuide 685, [...] thus.

Seing there is no fraction, [...] but a whole nūber, being 5. I double that, and than muste the number of y^e places be 10 Now from y^e last I subtracte the first, as 19 frō 118, thus: And so remaineth 99.

This 99 I diuide by a number lesse by 1 than the number of the places, and seing the places were 10, I diuide 99 [...] by 9, thus.

The quotient is 11, and so was the excesse, if I haue followed your rule right.

You haue wrought euery part of this question both well in order and truely in the practise of your rules.

I will than set it downe also for­mablie, so that the number of the places, the excesse and the total sum may streighte ap­peare, as your first example stoode.

The cōmon excesse. The Pro­gression.

That the places be 10, and that from [Page] the first to the last the commō excesse is 11, I perceiue most euidently, but whether the total summe be 685, I haue not yet proued, which I will now doe. I adds 19 and 118 togither that maketh 137: I multiplie that by halfe the number of the pla­ces [...] thus.

All thinges agrée most ex­actlye, so that I am perfecte y­nough in these rules, if I forget thē not a­gayne.

Vse maketh all things perfect.6

Your sixt rule is this. By the number of the places diuide the totall summe, double the quotient, and that will be the first and the last ioined in one sum. Than by a num­ber lesse by 1, than the number of the places multiplie the excesse, that ofcome subtracte from the first doubled quotient, and the half of the residue is the first number. The laste number you may diuerslye finde out, as by the first of our sixe rules, or by subtracting this first number from the sum which here contained both the firste and the last ioynt­lye (or thirdly) by continuall adding the ex­cesse.

I pray you make this somewhat [Page] more plaine with an example.

If euery moneth in the yeare (coun­ting them now as thirtéene) you gayned clearely 40 shillinges more than you didde the moneth next going before, and at the yeares ende you find the whole gaine 5720 shillinges, but yée remember not howe much either the gaine of the first moneth, or the last was, by this rule it maye be tried out.

So that here ye séeme to applie the 13 moneths to 13 places y^e 40 shillings euery one more then the other next before it, to be the common excesse, and 5700 s. to be the totall summe.

It is true: by 13, [...] then I diuide 5710, in this maner.

I double this quotient, so haue I 880 for the firste, and the last sinne ioyned togither, by 12, which is lesse by one than the [...] number of the places, I multi­plie 40, (the common excesse) so commeth 480.

This 480 I subtract from 880, so remaineth 400: halfe whereof [Page] is the firste number whiche we desired to know, that is 200.

And as for the last number, I can giue you it 3 wayes: As by the first of my sire rules, I multiplie the excesse by a number lesse by 1 than the number of the places: as 40 by 12, that giueth 480, whiche I adde to the first being 200, so shall the laste bée 680.

The same sum commeth forth, if yée sub­tract 200 from 880.

And thirdly, if I beginne 200, and so pro­céede, encreasing by 40, I shall at the thir­téenth place haue 680, as thus.

I thank you most hartilye for these 6 rules. Now if it be your pleasure, I would heare and learne somwhat of Progression Geometricall.

There are yet very many rules and propositions, which fall into this Arithme­ticall progression: but these shall suffise for this time.

[Page]And in Geometricall progression I will be more briefe, both because I haue bene so long in this part of Arithmeticall progres­sion, and also for that it woulde require the knowledge of Rootes, and numbers surde, (whereof ye haue yet learned nothing) if I shoulde frame the like propositions in them as I haue done in these. Therefore I will only teache you two practises, aboute it, and so ende the considerations & works of these progressions.Progression Geometri­call. Progression Geometricall is when the numbers increase by a like pro­portion, that is, if the second number con­taine the first, 2, 3, or 4 times, and so foorth: then the third containeth the seconde so ma­nye times also: and so the fourth the third, and the fifth the fourth: [...] wherefore I sette these thrée examples.

Here in the firste ex­ample you see, that e­uery number containeth the other (that go­eth next before him) 2 times: and in the se­conde example 3 times in the thirde exam­ple 5 times. Now if you will know how to finde easilye the summe of any suche num­bers, do thus. Consider by what nūber thoy [Page] bée multiplyed, whether by 2, 3, 4, 5, or a­nye other, and by the same number doe you multiplie the last summe in the Progressi­on.

I pray you worke it by this example, 2, 8, 32, 128, 512, 2048, whiche I haue fra­med by procéeding from 2, and continually multiplying by 4.

Then must I multiplie the last sum (which is 2048) by 4 also, and it will bée 8192. Nowe must I bate from this summe the first number of the progression, whiche here is 2, then resteth 8190, which summe I must diuide by 1 lesse then was the num­ber that I multiplied by. Seing then I multiplied by 4, I must diuide by 3, so diui­ding 8190 by 3, the quotient will be 2730, which is the summe of all the Progression. And now to proue whether you can doe the same, I giue you these numbers to adde by this rule, 3, 15, 75, 375, 1875, 9375, 46875.

I cannot well tell by what number this Progression doth increase.

In any such doubte, doe thus: Diuide the seconde number by the first,☜ and the quotiente will shewe you the number [Page] that engendreth the Progression.

Then is that number in this exam­ple 5, for so many times is 3 in 15.

So is it. Nowe worke as I taught you.

The laste number is 46875, whiche I multiplie be 5, and it yéeldeth 234375, from which I abate the first num­ber of the Progression, that is 3, and there resteth 234372, which I diuide by 4, for y^t is one lesse thē 5, & the quotient is 58593, which is the whole summe of the progressi­on.

Now that you know the summing of Geometricall Progression, I will shewe you a compendious manner eyther to pro­céede by, or to finde out the quantitye of a number whose distance from the first may be very greate,An abridge­ment in progression. whiche to doe by continu­all multiplication, would be very tedious, if the numbers be great and the places ma­ny.

Nothing can pleasure me more then breuitie, if it be playne.

I thinke I am not yet in a­nye point so darke or hard, that you néede to feare any obscuritie now. The manner is [Page] this: set downe of your progression foure or fiue of the first places, and vnder the first put a cipher, vnder the seconde 1, vnder the third 2, &c. as if yee had a progression encreasing by a fiue folde quantitie: as here, 2, 10, 50, 250, 1250: then vnder 2 I put a cipher, and vnder 10 the figure of [...] 4: and so foorthe if yée will: but to a wise and warie worker, a fewe places were sufficient to procéed by to any number of places in this sorte, if anye two of your numbers progres­sional be multiplied the one by the other, and the ofcome diuided by the firste of your progression, the quotient is one of your num­bers progressionall, and belonging to y^e place of your vnder numbers, that is equal to that summe, that is made of Addition togither of your two numbers which stoode vnder these two of your Progressionall numbers; that were multiplied the one by the o­ther, [...] as in thys example.

If I multiplie 10 by 50. thereof commeth [Page] 300, whiche I diuide by 2, (the firste number of the Progression,) and the quotient is 250: whiche 250, muste stande in the thirde place, because the number whiche standeth vnder 10 is 1, and that vnder 50. is 2: and 2 and 1 maketh 3. Therefore I say, that 250 belon­geth to the thirde [...] place of this pro­gression, as yée sée also here it doeth. Moreo­uer If I multi­plie 50, into it selfe, thereof commeth 2500: that 2500 I diuide by 2, the Quotiente is 1500, whiche muste be sette in the fourthe place: bicause 2 added to himselfe againe, ma­keth 4, and in our example 1250 occupieth the fourth place.

Then for that fifth place I multiplie the Progressionall numbers ouer 2 and 3 one by the other: and for the sixt, I multiplie that o­uer 3 in it selfe. &c.

Yée must wel remēber that these places that we nowe speake of, belong to the vnder numbers, for the true places of the vpper nū ­bers is euer one place more.

That I sée the reason of, by­cause [Page] the vnder numbers begin one after, and againste the first place of my progression stā ­deth a cypher, so that the 250 which you saide before did belong to the third place, I sée be­longeth to the number of 3 among your vn­der numbers, but from the true progressions beginning, it is the fourth.

You vnderstand me as I meane. Therefore for your exercise of both the rules here giuen for Geometricall progression, I will aske you a question, muche vsed among the common people, (as they haue a greate many the like) If I woulde sell you a Horse, hauing 4 shoes, and in euerie shoe six nailes, with this condition, that you shal pay for the first naile 1 ob, for the second 2 ob, for y^e thirde 4, and for the fourth 8, and so forth dubbling vntil the last naile. Now I demaunde of you how muche the price of the Horse woulde a­mount vnto?

Seing the Horse hathe 4 shoes, and in euerie shoe 6 nayles, I perceyue here wil bée 24 places. If I coulde nowe haue the laste number, I woulde quicklie dispatch this question. I wil therefore with as fewe multiplicatiōs as I can deuise, to come to the [Page] knowledge of the laste number of thys pro­gression. In double [...] I set foorth than a few of my progres­sion, thus.

If I nowe multiplie the numbers ouer 5 and 6, the one by the other, I shall haue the number of the eleuenth place for the vnder numbers, but of the twelfth for the vpper nū ­bers in which my progression standeth, and then that of y^e eleuenth place vnder, if I mul­tiplie in it selfe, I shal haue for the 22 place vnder, but for the 23 of that aboue, whiche I multiplie by that ouer 1 of my nether places, and I shall haue the 23 of my nether places, and the 24 of the vpper, which is the number I séeke for.

Me thinketh you haue forgotten youre rule for abridging your multiplicati­ons: for in it, the ofcome ouer of anye multi­plication, is to be diuided by the firste of the progression. And you now speake of no diui­sion.

Sir I néede not, as my progressiō beginneth now: for if I should diuide by 1, it maketh no other quotient, then the number is, it doth diuide.

It is verie wel remem­bred [...] and noted of you, to youre worke then according to youre prescribed manner, which I like well.

I multiply 64 by 32 as here. And it maketh 2048. whiche is the eleuenth place vnder, but the twelfth [...] aboue, and this, I multiply in it selfe, in this manner. And this is the 22 place vnder but the 23 aboue. I multiplie this then by 2, as héere.

And this of come [...] 8388608, is my foure and twentith place, whiche I haue found now by 3 multiplications.

Then doe I resort to the rule of summing this Progression, where I consider that the encrease of this summe procéedeth by multi­plication of 2, and therefore I doe multiplie the laste summe by 2 also, and it yéeldeth 16777216. from whiche 1 I abate the firste nūber which is 1, and then resteth 1677715, whiche I shoulde diuide by 1 lesse then I did [Page] [...] multiplie: but séeing that it is 1, I néede not to diuide it, for 1 (as I haue before saide) doth neither multiply nor diuide, therefore I take that summe 16777215 for the whole summe of the pence, which by Reduction I finde to be 699050 s, and 7 d, ob: that is 34952 lb, 10 s, 7 d, ob.

That is well done, but I thinke you wil buy no horse of the price.

No sir if I bée wise. Yet for my assurance wil I take so muche paine, as to come to this laste 8388608 by cōtinual mul­tiplication by 2, as in thys example you maye beholde my worke till I haue done.

Well, are you not almost weary?

Well fare my shorte rule, for introth it hathe more cunning and more [Page] ease.

Well, then answeare me to this question.

A Lorde deliuered to a Bricklayer a certayne number of loades of bricke, wher­of he willed him to make 12. walles, of such sorte, that the first wall shoulde receiue 2 thirdeles of the whole number: and the se­conde 2 thirdles of that that was left. And so euerie other 2 thirdles of that that remained: and so did the bricklayer: And when the 12 walles, were made, there remaineth one lode of Bricke.

Nowe I aske you, howe many loade went to euery wall, and howe many loade was in the whole?

Why sir, it is impossible for me to tel.

Nay, it is very easie, if you marke it well. Marke well that I saide, that euery wall shoulde receiue [...] thirdeles of the sum that was lefte. Nowe take away 2 thir­dels from anye summe and you must néedes graunt that that whiche remaineth is 1 thir­dle of the summe laste before: example of [...] ▪ from which if you take 2 thirdels, there will remaine 3, whiche is one thirdel of 9. Like­waies [Page] from 3 bate 2 thirdels: and there will remaine 1.

This is true, and nowe I per­ceiue, that the leaste wall had but two loade of bricke.

And by the same reason maye you knowe howe manie loade euerie wall hadde, according as this figure folowing doth shew, and likewayes what the whole summe of briekes was: for if you make 1 [...] summes, multiplying by 2, stil frō the last remayner, as you maye sée here on the lefte side of the table, there will appeare all the remayners of euerie wall: and if you multiplie the last of those 12 summes by [...], also, then will that be the sum of the loades whiche were deliue­red to the bricklayer.

Againe, if you doe double euery remainer, as you maye sée at the right side of this table, those numbers will showe the sum of loades that went to eache wall: whereby you maye perceiue, that each wall was 3 times so great as the next lesser.

Loe, nowe it appeareth easie ynoughe. Now surely I sée that Arithmetike is a right excellent arte.

You will say so when you know more [Page] of the vse of it: For this is nothing in com­parison to other points that may be wrought by it.

Then I beséech you sir, cease not to instruct me further in this wonderful cun­ning.

THE GOLDEN Rule.

BY order of the science (as men haue taughte it) there should fellow nexte the extraction of Rootes of number, whiche because it is somewhat harde for you, yet I will let it passe for a while, and will teache you the feate of the rule of Proportions, whiche for his ex­cellencie is called the Golden rule. Whose vse is, by thrée numbers knowen, to finde out an other vnknowen, whiche you desire to knowe: as thus. If you pay for your boorde for thrée moneths 16 shillings, howe muche shal you pay for 8 moneths.

To know this and all such like questions, you shall consider which two of your 3 num­bers be of one denominatiō, and set those two the one ouer the other, so that the vndermoste [Page] be it that the question is asked of: as in my question 3 and 8 be both of one denominati­on, for they both be monethes, and because 8 is the number that the question is asked of, I set them one ouer the other, [...] and 8 vndermoste, thus, with suche a crooked draught of lines. Then doe I set the other number whiche is 16, a­gainste 3, at the right side of [...] the line, thus.

And nowe to knowe my question, thys must I doe: I muste multiplie the lowermost on the left side, by that on the right side, and the summe that amounteth I muste diuide by the highest, on the left side. Or in playner wordes thus: I shall multiplie the number of whiche the question is asked (whyche is called the Thirde number) by the number of an other denomination,The third number. The seconde number. The first number. (whiche is cal­led the Seconde) and that summe that amoū ­teth muste I diuide by the summe of lyke denomination, whiche is called the Firste. Then for the knowledge of this question, I multiplie 8 into 16, and there amoūteth 128, whiche I diuide by 3, and it yéeldeth 42 shil­lings, and 2 s remaineth, whiche I turne into pennies, and they be 24 d, of whiche [Page] the third part is 8 d, so the third part of 128 s is 42 s, 8 d: which summe I write at the right hande of the figure a­gainst [...] 8, thus.

Hereby I knowe, that if thée monethes boording doe come to 16 s, that 8 monethes bording wil come to 42 s, 8 d: and likewise of any other like question.

But here must you marke, that the firste number and the thirde be of one denominati­on, and also the seconde and the fourth, the whiche you séeke: or else be of suche denomi­nations, that you in working may bring thē into one. As if a man shoulde aske me thys question.

*Twelue wéekes iournying coste me four­téene French Crownes at 6 s. the péece, how many poundes is that in one yeare. Here you sée no two nūbers of one denomination, But yet in working, you may turne them in­to like denominations, as thus: turne the one yeare into 52 weekes, and the fourth summe wil be French Crownes, by the order of the working: Then to knowe this question, multiplie the thirde summe 52 by the second 14: and the summe will be 728: that diuide by your firste number 12: and the quotiente [Page] will be 60. Crownes: And 8 Crownes re­mayning: whiche if you turne into shillings they will be 48 shillings which if you doe di­uide by your firste number 12 the quotiente will be 4: whiche signifyeth 4 s. put these 60. French Crownes (which make 18 poundes) with the 4 shillings: for [...] the summe that answe­reth to the question: And it is the iust expē ­ces of a yeare: And the summe wil be thus.

And take this euermore for a general Rule touching this whole Arte, That the doubtful or vnknowen number, that you woulde be resolued of, shall alwayes be set in the thirde place, note also the first number and the third, must euer be of one nature and denominati­on, or else must in working be brought to like denomination and then of necessitie must the other number be in the second place.

Remember also, that the place of the firste number is the highest on the left side: and the place of the seconde right against it on y^e right side: the place of the thirde number is vnder the firste, as by those examples you haue séene.

This I trust I can do.

But and if the question be asked thus: In 8 wéekes I spend 40. s howe long wil 105 shillings serue me? Here you sée that 8 wéekes aunsweres himselfe, and saieth 40 shillings. But how long time 105 shillings wil serue, you know not. Therefore you shal set 105 in the thirde place, according as I tolde you euen now. And the first place must alwayes be of the same nature or Denomi­nation, that the third is of, which here is 40. Then must 8 néeds be that other. Now mul­tiply 105 by 8 and it will be 840 which if you diuide by 40, it will yéelde 21, whiche is the Fourth number, and sheweth howe manye wéekes 105 s. will serue, if you spende 40 s. in eight wéekes.

The figure of thys que­stion [...] is this: as if you should saye: If 40 s. serue for 8 wéekes, 105 serue for 21 wéekes.

Other diuersities there be of working by this rule, but I hadde rather that you woulde learne this one well, than at the beginning to trouble your minde with many formes of working, sith this way can doe as much as al [Page] the other, and hereafter you shall learne the o­ther more conuenientlie.

And for your further aide and instruction to make you better acquainted wyth thys Goldon Rule, I haue here proponed 6 que­stions, and their aunsweres, whiche I thinke moste conueniente and méete to preferre the desirous to perfecte vnderstanding. The firste foure are all braunches of one Question sproong out of the beste trée, (for a young learner to taste of) that groweth in this Grounde of Artes, for that no manner of Question in the Rule of 3 what so euer it bée, can be proponed, but it muste be com­prehended, vnder the reason or style of one of these foure.

ADDITION.

THe easiest waye in this art, is to ad but two sums at once togither: howbeit, you may add more, as I wil tel you a­non. Therefore whē you wil add 2 sums you shall first set down one of thē, it forceth not which, and then by it draw a line crosse the other lines. And afterward set down the other sum, so that that line may be between them:Addition of 2 summes. as if you woulde

add 2659 to 8342, you must set your sums as you sée here.

And then if you list, you maye adde the one [Page] to the other in the same place: or else you may ad them both togither in a new place: which way, because it is most playnest, I will shew you first.

Therefore will I beginne at the vnits, which in the first sume is but 2, and in the second summe 9, that maketh 11. Those do I take vp, and for them I sette 11 in the new roome, thus.

Then doe I take vp all the Articles vn­der a hundred, which in the first summe are 40, and in the second summe 50, that ma­keth 90: or you may say better, that in the first summe there are 4 articles of 10, and in the seconde summe 5, whiche maketh 9, but then take héede that you set thē in their [Page] right lines as you sée here.

Where I haue taken away 40 from the firste summe, and 50 from the seconde, and in their stéede I haue sette 90 in the thirde roome, which I haue set plainelye that you might well perceiue it: how be it, seeing that 90 with the 10 that was in the thirde roome already, doth make

100, I might better for those 6 Counters set 1 in the thirde line, thus.

For it is all in one sum as you maye sée, but it is best neuer to sette fyue counters in any line, for that maye be done with one Counter in a higher place.

I iudge that good reason, for manye are vnnéedefull where one will serue.

Well, then will I adde forth of hundreds: I finde 3 in the first summe, and 6 in the second which maketh 900, them do I take vp, and set in the third roome, where is one hundred alreadie, to whiche I putte 900 and it will be 1000, therfore I set one counter in the fourth line for them all, as you sée here.

Then adde I the thousandes together, which in the first summe are 8000, and in the second 2000, that maketh 10000: them doe I take vp from those two places, and for them I set one counter in the fifte line, and then appeareth as you sée to be 11001, [Page]

for so manye doth a­mount of the Addition of 8342 to 2659.

Syr,To ad 8 summes to­gither. this I doe perceiue: but howe shall I sette one sūme to an other, not chaunging them to a third place?

Marke well how I doe it: I will adde togither 65436 and 3245, which firste I set down thus.

Then do I begin with the smallest, which in the first sūme is 5, that do I take vp, and would put to the other 5 in the second sum, [Page] sauing that two Counters cannot be set in a voyd place of 5 but for them both I must set 1 in the second lyne, which is the place of 10: therefore I take vp the fiue of the first summe, and the 5 of the second, and for them I set 1 in the secounde line, as you see here.

Then doe I likewise take vp the 4 coun­ters of the first sum and second line (which make 40) and ad them to the 4 counters of the same line, in the seconde summe, and it maketh 80, but as I sayde, I maye not conuenientlye set aboue 4 counters in one line, therefore to those 4 that I tooke vp in the first summe, I take one also of the se­cond [Page] summe, and then haue I taken vp 50, for which 5 Counters I sette down one in the space ouer the second line, as here doth appeare.

And then is there 80, as well with those 4 counters, as if you had set downe the other 4 also.

Now do I take the 200 in the first sum, and adde them to the 40 in the second sum, and it maketh 600, therefore I take vp the 2 counters in the first summe, and 3 of thē in the second summe, and for them 5, I set 1 in the space aboue, thus.

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Then I take the 3000 in the first sūme, vnto which there are none in the seconde summe agréeing, therefore I doe onelye remoue those thrée Counters from the first sum into the seconde, as here doth ap­peare.

And so you sée the whole summe that amounteth of the Addition of 65416 with 3245, to bée 68681.

[Page]And if you haue marked these two exam­ples well, you neede no further instruction in Addition of 2 onely summes: but if you haue more than two summes to adde, you may ad them thus.

Firste adde two of them, and then adde the thirde and the fourth, or more if there be so many: as if I woulde adde 2679 with 4286 and 1391. First I adde the two first summes thus.

And then I adde the third therto thus.

And so of more, if you haue them.

Nowe I thinke beste that you passe [Page] forthe to Subtraction, excepte there be anye wayes to examine this manner of Addition, then I thinke that were good to be knowen next.

A prooef.There is the same proofe here that is in the other Addition by the penne, I meane Subtraction, for that onelie is a sure waye: but considering that Subtraction muste be first knowen, I will first teach you the arte of Subtraction, and that by this ex­ample.

SVBTRACTION.

I Woulde subtracte 2892 out of 8746. These summes muste I sette downe as I did in Addition: but here it is beste to sette the lesser number first thus.

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Then shal I beginne to subtract the grea­test numbers first (contrarie to the vse of the pen) that is the thousandes in this example: therefore I finde amongst the thousands 2, for which I withdraw so manye from the se­cond summe (where are 8) and so remayneth there 6, as this example sheweth.

[Page]Then do I likewayes with the hundreds, of whiche in the firste summe I finde 8, and in the seconde summe but 7, out of whyche I can not take 8, therefore this must I doe: I muste looke howe muche my summe diffe­reth from 10, whiche I finde here to be 2, then muste I bate for my summe of 800, one thousande, and set downe the excesse of hun­dreds, that is to saye, 2, for so muche 1000 is more than I shoulde take vp. Therefore from the firste summe I take that 800, and from the seconde summe (where are 6000) I take vppe one thousande, and leaue 5000, but then set I downe the 200, vnto the 700 that are there alreadie, and make them 900, thus.

Then come I to the articles of tennes, where in the firste summe I finde 90, and in [Page] the seconde summe but onely 40. Now consi­dering that 90 can be bated from 40, I looke how muche that 90 doth differ from the next summe aboue it, that is 100, or else (which is all to one effecte) I looke howe much 9 doeth differ from 10, and I finde it to be 1, then in the steade of that 90, I doe take from the seconde summe 100: but considering that is 10 too muche, I set downe 1 in the nexte line beneth for it, as you sée

here.

Sauing that here I haue set one counter in the space, in steade of 5, in the next line.

And thus haue I subtract al, saue two whiche I muste bate from the 6 in the se­cond

summe and there will remaine 4, thus.

So that if I subtract 2892 from 8746, the remainer wil be 5854

And that this is tru­lie wrought you may proue by addition: for if you adde to this remainer the same summe [Page] that you did subtracte, then will the former summe 8746, amount againe.

That wil I proue: and first I set the sum that was Subtracted, which was 2892, and then the remainer 5854, thus.

Then doe I adde the first 2 to 4, which ma­keth 6: so take I vp 5 of those counters, and in their stead I set 1 in the space, and 1 in the lowest line, as here appeareth.

[Page]Then doe I adde the 90 next aboue to the 50, and it maketh 140, therefore I take vppe those 6 counters, and for them I set 1, to the hundreds in the thirde line, and foure in the second line thus.

Then do I come to the hundreds, of which I finde 8 in the first sum, & 9 in the seconde, y^t maketh 1700: therefore I take vp those 9 counters, & in their steade, I set 1 in y^e fourth line, and 1 in the space next beneath, and 2 in the third line as you sée here.

[Page]Then is there lefte in the firste summe but onelie 2000, whiche I shall take vppe from thence, and set in the same line in the

seconde summe, to the one that is there al­readie: and then wyll the whole summe ap­peare as you maye well sée, to be 8746, whiche was the firste grosse summe, and therefore I do perceiue that I had wel subtrac­ted before.

And thus you may sée, howe Subtraction may be tried by Addition.

I perceiue the same order here wyth Counters, that I learned before in figures.

Then let me sée how you can trie Ad­dition by Subtraction.

Firste I will sette forth thys ex­ample of Additiō, where I haue added 2189, to 4988. And the whole summe appeareth to be 7177.

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Nowe to trie whether that summe be wel added or no, I wil subtract one of the first two summes from the thirde, and if I haue well done, y^e remainer wil be like that other sum, as for example. I wil subtracte the first sum from the thirde, which I set thus in their or­der.

[Page]Then doe I subtract 2000 of the firste sum from the second summe, and then remaineth there 5000, thus.

Then in the third line I subtracte the 100 of the firste sum from the seconde sum where is onely 100 also: and thē in the third line resteth nothing, as you maye sée in the example following.

Then the in seconde line with his space o­uer him, I finde 80, whiche I shoulde sub­tract from the other sū then séeing ther are but onelie 70, I must take it out of some higher sum, which is here only 5000: therefore I take vp 5000: and séeing that is too much by 4920, I set down so many in the second roome, which with the 70 being there alreadie, do make 4990, and then the summes do stand thus.

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Yet remaineth there in the firste sum, 9 to be abated from the second sum, wherein that place of vnits doth appeare only 7: then must I bate a higher summe, that is to saye 10, but séeing that 10 is more than 9 (whiche I shoulde abate) by 1, therefore shall I take vp one counter from the

second, and set down the same in the firste or lowest line, as you see here.

And so haue I en­ded this worke, and the summe appea­reth to be the same whyche was the se­conde summe of mine Addition, and therefore I perceiue I [Page] haue wel done.

To stande longer about this, it is but folly, [...]nother [...]ay of ad­ [...]tion. except that this you may also vnderstād, that many do begin to subtract with counters not at the highest summe as I haue taughte you, but at the neathermost, as they do vse to adds: and when the summe to be abated in a­nye line appeareth greater than the other, thē do they borrow one of the next higher roome, as for example.

If I shoulde abate 1846 from 2378, they set the summes thus.

First they take 6, whiche is the lower line, and his space, from 8 in the same roomes in the seconde summe, and yet there remay­neth 2 counters in the lowest line. Then in the seconde line must 4 be subtracted from 7, and so remaineth there 3. Then 800 in the thirde line, and his space, from 300 of the se­conde summe can not be, therefore doe they bate it frō a higher roome, that is from 1000: and because that 1000 is too muche by [...]0 [...], therfore must I set downe 200 in the thirde line, after I haue taken vp 1000 from the [Page] fourth line. Then is there yet 1000 in the fourth line of the first sum, whiche if I with­drawe from the seconde sum, then doth al the figures stand in order, thus.

So that (as you

sée) it differeth not greatly whe­ther you beginne subtraction at the higher lines, or at the lower.

Howe be it, as some men like y^e one way beste, so some like the other: therefore you nowe knowing bothe, maye vse whyche you liste.

MVLTIPLICA­TION.

BVt nowe touchyng Multiplication: you shall sette your num­bers in two roomes (as you did in those other kindes) but 2 so y^t the multiplier be set in the first roome, then shall you begin with the highest numbers of the seconde roome, and multiplie them firste, after this sort. Take y^e ouermost line in your first working,☞ as if it were the lowest line, setting on it some moueable marke (as you list) and looke how many counters be in him, take them vp, and for them sette downe the whole multiplier so many times as you tooke vp counters: reckning (I say) that line for the Vnites. And when you haue done with the highest number, then come to the nexte lyne beneath, and do euen so with it, and so with the nexte, till you haue done all. And if there be anye number in a space, then for it shall [Page] you take the multiplier 5 times: and then muste you recken that line for the Vnites, whiche is next beneath that space. Or else after a shorter waye, you shall take onelie halfe the multiplier, but thē shal you take the line next aboue that space for the line of Vnites. But in suche working, if by chaunce your multi­rlyer be an odde number, so that you can not take the halfe of it iustlye, then muste you take the greater halfe, and set downe that, as if that it were the iuste halfe: and further you shall sette one Counter in the space beneath that line, which you recken for the line of V­nits, or else onely remoue forwarde the same that is to be multiplied.

If you set forth an exāple hereto, I think I shal perceiue you.

Take

this example: I woulde multi­plie 1 5 4 2 by 2 6 5, therfore I set the numbers thus.

[Page]Then firste I beginne at the 1000 in the highest roome, as if it were the first place, and I take it vp, setting downe for it so often (that is once) the multiplyer, which is 365, thus as you sée here: where, for the one counter ta­ken vp from the fourth line, I haue set down other 6, which make y^e sum of the multiplier, reckening that fourth line as if it were the first, which thing I haue marked by the hand set at the beginning of the same.

I perceiue this well, for in déede this sum that you haue set down is 265000: for so much doth amounte of 1000, multiplied by 365.

Well, then to goe foorth, in the [Page] next space I finde one counter, whiche I re­moue forward, but take it not vp, but doe (as in such case I must) set down the greater half of my multiplier (séeing it is an od number) which is 182, and here I doe still lette that fourth place stand, as if it were the first: as in these examples you shall sée.

☜ Where I haue set this multiplication with other, but for the ease of your vnderstāding, I haue set a little line betwéene them. Now should they both in one summe stand thus.

☜

[Page] An other forme of Multiplica­tion.

Howbeit, an other fourme to multiplye such coūters in space, is this: Firste to re­moue y^e finger to the line next beneth that space, and thē to take vp that Counter, and to set down the mul­tiplier fiue times: as here you see.

Which summes if you do adde togither into one summe, you shall perceiue that it will be the same that appeareth of the o­ther working before, so that both sorts are to one intente: but as the other is shor­ter, so this is play­ner to reason for such as haue had smal ex­ercise in this arte. Notwithstāding you [Page] may adde them in your minde before you set them downe: as in this example you might haue sayde, 5 times 300 is 1500, and 5 times 60 is [...]00 also 5 times 5 is 25, which all put togither, doe make 1825, whiche you may at one time set downe if you list.

But now to go forth, I must remoue the hande to the next counters whiche are in the second lyne, and there must I take vp those 4 counters setting downe for them my mul­tiplier 4 times seuerally, or else I maye ga­ther that whole sum in my mynde firste, and then set it downe: as to say, 4 times 300 is 1200: 4 times 60 are 240: and 4 times 5 make 20, that is in all 1460, that shall I set downe also, as here you sée.

Which if I ioyne in one sum with the for­mer [Page] numbers it will appeare thus,

Then to ende this Multiplication, I re­moue the finger to the lowest line, where are only 2, them do I take vp, and in their stéede doe I set downe twice 365, that is 730, for which I set one in the space aboue the thirde line for 500, and 2 more in the thirde lyne with that one that is there alreadie, and the rest in their order, and so haue I ended the whole summe, thus.

[Page]Whereby you sée, that 1542 (whiche is the number of yeares sith Christe his incarnati­on) being multiplyed by 365 (which is the number of dayes in one yeare) doth amounte vnto 562830,The summe of the daies sith Christs incarnation which declareth the nūber of dayes sith Christes incarnation vnto the end of 1542 yeares, (beside 385 dayes and 12 houres for leape yeares.)

Now will I prooue by an other exam­ple, as this: 40 labourers (after 6 d the day for eche man) haue wroughte 28 dayes: I would know what their wages doth amoūt vnto.

In this case must I worke doublye: first I must multiplie the number of the labou­rers by the wages of a man for one daye, so will the charge of one day amount. Then secōdarily shal I multiplie the charge of one daye by the

whole number of dayes, and so wil y^e whole sum appear: first therfore I shal set the sūmes thus:

Where in the firste place is the Multiplier (that is 1 dayes wages for one man) & in the seconde space is [Page] set the nūber of y^e warkmē to be multiplied.

Then saye I: 6 times 4 (reckoning that second line of the line of Vnits) maketh 24, for which summe I

should set 2 counters in the third line, and 4 in the second there­fore doe I set 2 in the third line, and let the 4 stand still in the se­cond line thus.

So appeareth the whole dayes wages to be 240 d y^t is 20 s.

Then doe I multi­plye

agayne the same summe by the nūber of dayes, and firste I set the nūbers thus, Thē because ther are

counters in diuerse lines, I shall begin with the highest, and take them vp, setting for them the multi­plier so many times as I tooke vp coun­ters, that is twise, then wil y^e sum stād thus.

[Page]Then come I to the second line, and take vp those 4 Counters, setting for thē the mul­tiplier foure times, so

wil the whole summe appeare thus.

So is the whole wa­ges of 40 workemen for 28 days (after 6 d eche daye for a man) 6720 d that is 560 s or 28 pound.

Now if you would prooue Multipli­cation, the surest way is by Diuision: there­fore will I ouerpasse it, till I haue taughte you the arte of Diuision, whiche, you shall worke thus.

DIVISION.

FIrst set downe the diuisor, for feare of forgetting, and thē set the nūber that shal be diuided, at the right side, so farre from the Diuisor, that the quotient may be set betwéene them: as for example.

If 225 shéepe cost 45 lb. what did euerye [Page] shéepe cost? To know this, I shoulde diuide the whole summe that is 45 lb, by 225, but that cannot be: therefore must I firste reduce that 45 lb into a lesser denomination, as into shillinges, then I multiplie 45 by 20, and it is 900: that summe shall I diuide by the number of sheepe, which is 225, these two numbers therefore I set thus.

Then begin I at the highest lyne of the di­uident, and séeke how often I maye haue the diuisour therein, and that maye I doe foure times: then saye I, foure times 2 are 8, whi­the if I take from 9, there resteth but 1, thus.

[Page]And because I founde the diuisor 4 times in the diuident, I haue set as you sée, 4 in the middle roome, which is the place of the quotient: but now must I take the rest of y^e diuisour as often out of the remayner, there­fore come I to the seconde line of the diuisor, saying: 2 foure times make 8, take 8 from 10, and there resteth 2, thus.

Then come I to the lowest number which is 5, and multiplie it 4 times, so is it 20, that take I from 20, & there remayneth nothing,☜ so that I sée my quotient to be 4, which are in valewe shillings, for so was the diuident: and therby I know y^t if 225 Shéepe did cost 45 lb, euery shéepe cost 4 s.

This can I doe,Example of vvages as you shall per­ceiue by this exāple. If 100 soldiours do spēd euery moneth 68 lb, what spendeth ech man?

First because I cannot diuide the 68, by 160, therefore I will turne the lb into pen­nies [Page] by multiplicatiō, so shal there be 16320 d. Now must I diuide this summe by the number of souldiors, therefore I set them in order thus.

Then beginne I at the highest place of the diuidend, séeking my Diuisor there, which I finde once, therefore sette I 1 in the nether line.

Not in the nether line of the whole summe, but in the nether lyne of that worke which is the third line.

So standeth it with reason.

Then thus doe they stand.

[Page]Then séeke I agayne the rest, how often I may finde my diuisour: and I sée that in the 300 I mighte finde 100 thrée tymes, but then the 60 will not be so often found in [...]0, therefore I take 2 for my quotient: thē take I 100 twice frō 300, and there resteth 100, out of which with the 20 (that maketh 120) I may take 60 also twice, and then standeth the numbers thus.

Where I haue set the quotient 2 in the lowest line: So is euery Souldiors portion 102 d that is 8 s, 6 d.

But yet because you shall iustlye perceiue the reason of Diuision, it shall bée good that you doe set your diuisor stil agaynst those numbers from which you doe take it, as by this example I will declare.

If the purchase of 20 acres of ground did cost 290 pound, what did one acre cost?Example of purchase.

First will I turne the poundes into pen­nies, [Page] so will there be 69600 pence. Then in setting down these numbers, I shal do thus. First set the diuidend on the right hand as it ought, and then the diuisor on the lefte hande agaynst those numbers from which I intend to take him firste as here you sée, where I haue set the diuisor two lines higher than is his owne place.

This is like the order of Diuision by the pen.

Truth you say, and now must I sette the quotient of this worke in the thirde line, for that is the lyne of vnits in respecte to the diuisor in this worke.

Then I séeke how often the diuisor maye be found in the diuident, and that I fynd 3 tymes, then set I 3 in the third lyne for the quotient and take awaye that 60000 from the diuidend, and farther I do set the diuisor [Page] one line lower, as you see here.

And then séeke I howe often the diuisor will be taken from the number agaynste it, which will be 4 times and 1 remaining.

But what if it chaunce that when the diuisor is so remoued, it cannot be ones taken out of the diuident against it?

Then muste the diuisour be set in an other line lower.

So was it in diuision by the pen, and therefore was there a cipher set in the quoti­ent: but how shall that be noted here?

Here néedeth no token, for the lines doe represent the places: onely looke that you set your quotient in that place whiche stan­deth for vnits in respect of y^e diuisor: but now to returne to the example. I finde the diuisor 4 times in the diuident, and 1 remaining, for 4 times 2 make 8, which I take frō 9, & there [Page] resteth 1, as this figure following sheweth; and in the middle space for the quotient I set 4 in the second line [...] he is in this worke the place of vnits.

☞

Then remoue I the diuisour to the next lo­wer line, & seeke how often I may haue it in the diuident, which I may doe here 8 tymes iust, and nothing remain, as in this fourme.

Where you may sée, that the whole quoti­ent is 348 d, that is 29 s, whereby I knowe that so much cost the purchase of one acre.

Now resteth the proues of Multiplica­tion, and also of Diuision.

Their best proues are eche one by the other: for multiplication is prooued by Di­uision, and Diuision by Multiplication, as in the worke by the pen you learned.

If that be all, you shall not néede to re­peate agayn that that was sufficiētly taught alreadie: and except you will teache me any other feate, here maye you make an ende of this art, I suppose.

So will I doe as touching whole number: and as for brokē number, I will not trouble your wit with it, till you haue pra­ctised this so well, that you be full perfect, so that you néede not to doubt in any point that I haue taught you, and then maye I boldlye instruct you in the arte of Fractions or Bro­ken number: wherein I will also shew you the reasons of al that you haue now learned. But yet before I make an end, I will shew you the order of common casting, wherin are both pennies, shillings, and poundes, procée­ding by no grounded reason, but onelye by a receyued forme, and that diuerslye of diuerse men: for the Marchantes vse one forme, and Auditours an other.

Marchantes vse.

BVt first for Mar­chantes

fourme, marke this ex­ample here, in whiche I haue expressed this sum 198 lb 19 s 11 d. So that you maye sée that the lowest line serueth for pen­nies, the nexte aboue for shillings, the third for pounds, and the fourth for scores of poun­des.

And further you may sée, that the space be­twéene d and s may receiue but one counter (as al other spaces likewise do) and that one standeth in that place for 6 d.

Likewise betwéene the shillinges and the poundes, one counter standeth for 10 s.

And betwéene the poundes and 20 lb. one counter standeth for 10 lb.

But beside those you maye sée at the lefte side of shillinges, that one counter standeth alone, and betokeneth 5 s.

So agaynst the poundes, that one counter standeth for 5 lb. And against the 20 pounds, the one counter standeth for 5 score pounds, that is 100 pounde, so that euery side coūter [Page] is 5 times so muche as one of them againste which he standeth.

THE ARTE OF NVM­bring on the hande.

BVt one feate I shall teach you, which not onely for y^e strange­nesse and secretenesse is muche pleasant, but also for the good commoditie of it, right worthy to be wel marked.

This feate hath bin vsed aboue 2000 yeres at the least, and yet was it neuer commonlie knowen, especiallie in Englishe it was neuer taught yet. This is the arte of numbring on the hand, with diuerse gestures of y^e fingers, expressing anye sum conceiued in the minde. And first to beginne.

If you will expresse anye sum vnder 100, you shall expresse it with your left hand and from 100 vnto 1000, you shal expresse it with your right hand, as here or­derly by this Table follo­wing you maye perceiue.

Here followeth the Table of the Art of the hande.

[Page]

[hand gesture]

1

[hand gesture]

2

[hand gesture]

3

[hand gesture]

4

[hand gesture]

5

[hand gesture]

6

[hand gesture]

7

[hand gesture]

8

[hand gesture]

9

[hand gesture]

10

[hand gesture]

20

[hand gesture]

30

[hand gesture]

40

[hand gesture]

50

[hand gesture]

60

[hand gesture]

70

[hand gesture]

80

[hand gesture]

90

[hand gesture]

100

[hand gesture]

200

[hand gesture]

300

[hand gesture]

400

[hand gesture]

500

[hand gesture]

600

[hand gesture]

700

[hand gesture]

800

[hand gesture]

900

[hand gesture]

1000

[hand gesture]

2000

[hand gesture]

3000

[hand gesture]

4000

[hand gesture]

5000

[hand gesture]

6000

[hand gesture]

7000

[hand gesture]

8000

[hand gesture]

9000

[Page] 1 1 In which (as you may sée) 1 is expressed by the little finger of the left hand, closely and harde crooked.

2 2 Is declared by like bowing of the wed­ding finger (which is the next to the little fin­ger) togither with the little finger.

3 3 Is signifyed by the middle finger, bowed in like manner with these two.

4 4 Is declared by the bowing of the middle finger, and the ring finger or wedding finger, with the other all stretched forth.

5 5 Is represented by the middle finger one­ly bowed.

6 And 6 by the wedding finger onlie crooked: and thus you may mark in these a certain or­der. But now 7, 8, and 9, are expressed wyth the bowing of the same fingers, as are 1, 2, & 3, but after another forme.

7 For 7 is declared by the bowing of y^e little finger as is 1, saue that for 1 the finger is clasped in, hard and round, but for to expresse 7, you shall bow the middle ioynt of the little finger onelye, and holde the other ioyntes straight.

If you wil giue me leaue to expresse it after my rude maner, thus I vnderstād your meaning: that one is expressed by crooking in [Page] the little finger, like the head of a Bishoppes bagle: and 7 is declared by the same finger bowed like a gibbet.

So I perceiue you vnderstand it.

Then to expresse 8, you shall bowe after 8 the same maner both the little finger, and the ring finger.

And if you bowe likewise with them the 9 middle finger, then doth it betoken 9.

Nowe to expresse 10, you shal bowe your 10 forefinger rounde, and set the end of it on the highest iointe of the thumbe.

And for to expresse 20, you muste set youre 20 fingers straight, and the end of your thumb to the partition of the formost & middle finger.

30 Is represented by the ioyning togither 30 of the heads of the foremost finger & the thūb.

40 Is declared by setting of the thumbe 40 crossewaies on the formost finger.

50 Is signified by right stretching foorth of 50 the fingers iointly and applying of y^e thumbs ende to the partition of the middle finger, and the ring finger or wedding finger.

60 Is formed by bending of the thumbe 60 crooked, and crossing it with the forefinger.

70 Is expressed by the bowing of the fore­most 70 finger and setting the ende of the thūbe [Page] betwéene the 2 formost or highest ioints of it.

80 80 Is expressed by setting of the foremoste finger crossewayes on the thumbe, so that 80 differeth thus frō 40: for that 80, the forefin­ger is set crossewayes on the thumbe and for 40 the thumb is set crosse ouer the forefinger.

90 90 Is signified by bending the forefinger, and setting the end of it in y^e innermost ioint of the thumbe, that is euen at the foote of it. And thus are al the nūbers ended vnder 1 [...]0.

In déed these be al he numbers from 1 to 10, & then all the tenthes within 100, but 11, 12, 13, 21, 22, 23, this teacheth me not how to expresse 11, 12, 13, &c. 21, 22, 23, &c. and such like.

You can little vnderstand, if you cā not doe that without teaching. What is 11? is it not 10 and 1? then expresse 10 as you were taught and 1 also, that is 11: and for 12 ex­presse 10 and 2: for 23 set 20 and [...]: and so for 68, you must make 6, and thereto 8: and so of al other sortes.

But now if you wold represent 100, either any number aboue it, you must doe that with the right hand, after this maner.

100 You must expresse 100 in the righte hande with the little finger, so bowed as you did ex­presse 1 in the left hand.

[Page]And as you expressed 2 in the lefte hande, the same fashion in the right hande doeth de­clare 200.200

The fourme of 3 in the right hand standeth for 300.300

The forme of 4 for 400.400

Likewise the forme of 5, for 500.500

The forme of 6, for 600. And to be shorte:600 looke how you did expresse single vnities and tenthes in the left hande, so must you expresse vnities and tenthes of hundreds, in the right hand.

I vnderstande you thus: that if I woulde represente 900, I muste so forme the 900 fingers of my right hande to expresse 9. And as in my lefte hande I expresse 10, so in my right hand must I expresse a 1000.1000

And so the forme of euerie tenth in the lefte hand, serueth to expresse the number of thou­sands, so the summe of 40 standeth for 4000.4000

The summe of 80, for 8000.8000

And the forme of 90 (which is the greatest) for 9000, and aboue that I can not expresse 9000 any number.

No, not with one finger, howe be it, with diuerse fingers you maye expresse 9999. and al at one time, & that lacketh but [Page] 1 of 10000. So that vnder ten thousande you maye by your fingers expresse anye summe. And this shall suffise for Numeration on the fingers. And as for Addition, Subtraction, Multiplication, and Diuision (whiche yet were neuer taught by any man as farre as I doe knowe) I will instruct you after the trea­tise of Fractions: and now for this time fare well, and looke that you ceasse not to practise that you haue learned.

Sir, with most hartie minde I thanke you, both for your good learning and also your good counsel, which (God willing) I truste to follow.