Chapter 8

Here folowith{e} the extraccio{u}n of rotis, and first in nombre q{ua}drat{es}. Wherfor me shall{e} se what is a nombre quadrat, and what is the rote of a nombre quadrat, and what it is to draw out the rote of a nombre. And before other note this divisio{u}n: Of nombres one is lyneal, anoþ{er} sup{er}ficiall{e}, anoþ{er} quadrat, anoþ{er} cubik{e} or hoole. lyneal is that þat is considred{e} after the p{ro}cesse, havyng{e} no respect to the direccio{u}n of nombre in nombre, As a lyne hath{e} but one dymensio{u}n that is to sey after the length{e}. Nombre sup{er}ficial is þ{a}t cometh{e} of ledyng{e} of oo nombre into a-nother, wherfor it is called{e} sup{er}ficial, for it hath{e} .2. nombres notyng or mesuryng{e} hym, as a sup{er}ficiall{e} thyng{e} hath{e} .2. dimensions, þ{a}t is to sey length{e} and brede. And for bycause a nombre may be had{e} in a-nother by .2. man{er}s, þ{a}t is to sey other in hym-self{e}, oþ{er} in anoþ{er}, Vnderstond{e} yf it be had in hym-self, It is a quadrat. ffor dyvisio{u}n write by vnytes, hath{e} .4. sides even as a quadrangill{e}. and yf the nombre be had{e} in a-noþ{er}, the nombre is sup{er}ficiel and not quadrat, as .2. had{e} in .3. maketh{e} .6. that is þe first nombre sup{er}ficiell{e}; wherfor it is open þat all{e} nombre quadrat is sup{er}ficiel, and not co{n}u{er}tid{e}. The rote of a nombre quadrat is þat nombre that is had of hym-self, as twies .2. makith{e} 4. and .4. is the first nombre quadrat, and 2. is his rote. 9. 8. 7. 6. 5. 4. 3. 2. 1. / The rote of the more quadrat .3. 1. 4. 2. 6. The most nombre quadrat 9. 8. 7. 5. 9. 3. 4. 7. 6. / the remenent ou{er} the quadrat .6. 0. 8. 4. 5. / The first caas of nombre quadrat .5. 4. 7. 5. 6. The rote .2. 3. 4. The second{e} caas .3. 8. 4. 5. The rote .6. 2. The third{e} caas .2. 8. 1. 9. The rote .5. 3. The .4. caas .3. 2. 1. The rote .1. 7. / The 5. caas .9. 1. 2. 0. 4. / The rote 3. 0. 2. The solid{e} nombre or cubik{e} is þat þ{a}t comytħe of double ledyng of nombre in nombre; And it is cleped{e} a solid{e} body that hath{e} þ{er}-in .3 [dimensions] þat is to sey, length{e}, brede, and thiknesse. so þ{a}t nombre hath{e} .3. nombres to be brought forth{e} in hym. But nombre may be had{e} twies in nombre, for other it is had{e} in hym-self{e}, oþ{er} in a-noþ{er}. If a nombre be had{e} twies in hym-self, oþ{er} ones in his quadrat, þ{a}t is the same, þ{a}t a cubik{e} [*Fol. 54.] is, And is the same that is solide. And yf a nombre twies be had{e} in a-noþ{er}, the nombre is cleped{e} solide and not cubik{e}, as twies .3. and þ{a}t .2. makith{e} .12. Wherfor it is opyn{e} that all{e} cubik{e} nombre is solid{e}, and not {con}u{er}tid{e}. Cubik{e} is þ{a}t nombre þat comyth{e} of ledyng{e} of hym-self{e} twyes, or ones in his quadrat. And here-by it is open that o nombre is the roote of a quadrat and of a cubik{e}. Natheles the same nombre is not q{ua}drat and cubik{e}. Opyn{e} it is also that all{e} nombres may be a rote to a q{ua}drat and cubik{e}, but not all{e} nombre quadrat or cubik{e}. Therfor sithen þe ledyng{e} of vnyte in hym-self ones or twies nought cometh{e} but vnytes, Seith{e} Boice in Arsemetrik{e}, that vnyte potencially is al nombre, and none in act. And vndirstond{e} wele also that betwix euery .2. quadrat{es} ther is a meene p{ro}porcionall{e}, That is opened{e} thus; lede the rote of o quadrat into the rote of the oþ{er} quadrat, and þan wolle þe meene shew.

[Sidenote: Examples of square roots.]

+-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+ | Residuu{m} | | |0| || | | |4|| | |0| | || | | 0 | | +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+ | Quadrand{e} |4|3|5|6||3|0|2|9||1|7|4|2|4||1| 9 | 3 |6| +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+ | Duplum |1|2| | ||1|0| | ||2| |6| | || |[8]|[{19}]| | +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+ | Subduplu{m} | |6| |6|| |5| |5||1| |3| |2|| | 4 | |4| +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+

[Sidenote: A note on mean proportionals.]

Also betwix the next .2. cubikis, me may fynde a double meene, that is to sey a more meene and a lesse. The more meene thus, as to bryng{e} the rote of the lesse into a quadrat of the more. The lesse thus, If the rote of the more be brought Into the quadrat of the lesse.

[Headnote: Chapter X. Extraction of Square Root.]

[Sidenote: To find a square root. Begin with the last odd place. Find the nearest square root of that number, subtract, double it, and set the double one to the right. Find the second figure by division. Multiply the double by the second figure, and add after it the square of the second figure, and subtract.]

[{20}]To draw a rote of the nombre quadrat it is What-eu{er} nombre be p{ro}posed{e} to fynde his rote and to se yf it be quadrat. And yf it be not quadrat the rote of the most quadrat fynde out, vnder the nombre p{ro}posed{e}. Therfor yf thow wilt the rote of any quadrat nombre draw out, write the nombre by his differences, and compt the nombre of the figures, and wete yf it be od{e} or even. And yf it be even, than most thow begynne worche vnder the last save one. And yf it be od{e} w{i}t{h} the last; and forto sey it shortly, al-weyes fro the last od{e} me shall{e} begynne. Therfor vnder the last in an od place sette, me most fynd{e} a digit, the which{e} lad{e} in hym-self{e} it puttith{e} away that, þat is ou{er} his hede, oþ{er} as neigh{e} as me may: suche a digit found{e} and w{i}t{h}draw fro his ou{er}er, me most double that digit and sette the double vnder the next figure toward{e} the right hond{e}, and his vnder double vnder hym. That done, than me most fy{n}d{e} a-noþ{er} digit vnder the next figure bifore the doubled{e}, the which{e} [*Fol. 54b] brought in double setteth{e} a-way all{e} that is ou{er} his hede as to reward{e} of the doubled{e}: Than brought into hym-self settith{e} all away in respect of hym-self, Other do it as nye as it may be do: other me may w{i}t{h}-draw the digit [{21}][last] found{e}, and lede hym in double or double hym, and after in hym-self{e}; Than Ioyne to-geder the p{ro}duccion{e} of them bothe, So that the first figure of the last p{ro}duct be added{e} before the first of the first p{ro}duct{es}, the second{e} of the first, {et}c. and so forth{e}, subtrahe fro the totall{e} nombre in respect of þe digit.

[Sidenote: Examples.]

+------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+ | The residue | | | | | || | | | | || | | |5| 4 |3|2| +------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+ | To be quadred{e} |4|1|2|0|9||1|5|1|3|9|| 9 |0| 0 |5| 4 |3|2| +------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+ | The double | |4|0| | || |2| |4| || |6| |0| | |0| +------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+ | The vnder double |2| |0| |3||1| |2| |3||[3]| |[0]| |[0]| |0| +------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+

[Sidenote: Special cases. The residue.]

And if it hap þ{a}t no digit may be found{e}, Than sette a cifre vndre a cifre, and cesse not till{e} thow fynde a digit; and whan thow hast founde it to double it, neþ{er} to sette the doubled{e} forward{e} nether the vnder doubled{e}, Till thow fynde vndre the first figure a digit, the which{e} lad{e} in all{e} double, settyng away all{e} that is ou{er} hym in respect of the doubled{e}: Than lede hym into hym-self{e}, and put a-way all{e} in regard{e} of hym, other as nygh{e} as thow maist. That done, other ought or nought wolle be the residue. If nought, than it shewith{e} that a nombre componed{e} was the quadrat, and his rote a digit last found{e} w{i}t{h} vnder{e}-double other vndirdoubles, so that it be sette be-fore: And yf ought[{22}] remayn{e}, that shew{i}t{h} that the nombre p{ro}posed{e} was not quadrat,[{23}] [[wher-vpon{e} se the table in the next side of the next leef{e}.]] but a digit [last found with the subduple or subduples is]

[Sidenote: This table is constructed for use in cube root sums, giving the value of ab.^2]

+---+-----+-----+-----+-----+-----+---------+-----+---------+ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | +---+-----+-----+-----+-----+-----+---------+-----+---------+ | 2 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | +---+-----+-----+-----+-----+-----+---------+-----+---------+ | 3 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | +---+-----+-----+-----+-----+-----+---------+-----+---------+ | 4 | 32 | 48 | 64 | 80 | 96 |112[{24}]| 128 | 144 | +---+-----+-----+-----+-----+-----+---------+-----+---------+ | 5 | 50 | 75 | 100 | 125 | 150 | 175 | 200 | 225 | +---+-----+-----+-----+-----+-----+---------+-----+---------+ | 6 | 72 | 108 | 144 | 180 | 216 | 252 | 288 | 324 | +---+-----+-----+-----+-----+-----+---------+-----+---------+ | 7 | 98 | 147 | 196 | 245 | 294 | 343 | 393 | 441 | +---+-----+-----+-----+-----+-----+---------+-----+---------+ | 8 | 128 | 192 | 256 | 320 | 384 | 448 | 512 | 576 | +---+-----+-----+-----+-----+-----+---------+-----+---------+ | 9 | 168 | 243 | 324 | 405 | 486 | 567 | 648 |729[{25}]| +---+-----+-----+-----+-----+-----+---------+-----+---------+

[Sidenote: How to prove the square root without or with a remainder.]

The rote of the most quadrat conteyned{e} vndre the nombre p{ro}posed{e}. Therfor yf thow wilt p{ro}ve yf thow have wele do or no, Multiplie the digit last found{e} w{i}t{h} the vnder-double oþ{er} vnder-doublis, and thow shalt fynde the same figures that thow haddest before; And so that nought be the [*Fol. 55.] residue. And yf thow have any residue, than w{i}t{h} the addicio{u}n þ{er}of that is res{er}ued{e} w{i}t{h}-out in thy table, thow shalt fynd{e} thi first figures as thow haddest them before, {et}c.

[Headnote: Chapter XI. Extraction of Cube Root.]

[Sidenote: Definition of a cubic number and a cube root. Mark off the places in threes. Find the first digit; treble it and place it under the next but one, and multiply by the digit. Then find the second digit. Multiply the first triplate and the second digit, twice by this digit. Subtract. Examples.]

Heere folowith{e} the extraccio{u}n of rotis in cubik{e} nombres; wher-for me most se what is a nombre cubik{e}, and what is his roote, And what is the extraccio{u}n of a rote. A nombre cubik{e} it is, as it is before declared{e}, that cometh{e} of ledyng of any nombre twies in hym-self{e}, other ones in his quadrat. The rote of a nombre cubik{e} is the nombre that is twies had{e} in hy{m}-self{e}, or ones in his quadrat. Wher-thurgh{e} it is open, that eu{er}y nombre quadrat or cubik{e} have the same rote, as it is seid{e} before. And forto draw out the rote of a cubik{e}, It is first to fynd{e} þe nombr{e} p{ro}posed{e} yf it be a cubik{e}; And yf it be not, than thow most make extraccio{u}n of his rote of the most cubik{e} vndre the nombre p{ro}posid{e} his rote found{e}. Therfor p{ro}posed{e} some nombre, whos cubical rote þ{o}u woldest draw out; First thow most compt the figures by fourthes, that is to sey in the place of thousand{es}; And vnder the last thousand{e} place, thow most fynde a digit, the which{e} lad{e} in hym-self cubikly puttith{e} a-way that þat is ou{er} his hede as in respect of hym, other as nygh{e} as thow maist. That done, thow most trebill{e} the digit, and that triplat is to be put vnder the .3. next figure toward{e} the right hond{e}, And the vnder-trebill{e} vnder the trebill{e}; Than me most fynd{e} a digit vndre the next figure bifore the triplat, the which{e} w{i}t{h} his vnder-trebill{e} had into a trebill{e}, aft{er}warde other vnder[trebille][{26}] had in his p{ro}duccio{u}n, putteth{e} a-way all{e} that is ou{er} it in regard{e} of[{27}] [the triplat. Then lade in hymself puttithe away that þat is over his hede as in respect of hym, other as nyghe as thou maist:] That done, thow most trebill{e} the digit ayene, and the triplat is to be sette vnder the next .3. figure as before, And the vnder-trebill{e} vnder the trebill{e}: and than most thow sette forward{e} the first triplat w{i}t{h} his vndre-trebill{e} by .2. differences. And than most thow fynde a digit vnder the next figure before the triplat, the which{e} with{e} his vnder-t{r}iplat had in his triplat afterward{e}, other vnder-treblis lad in p{ro}duct [*Fol. 55b] It sitteth{e} a-way ałł that is ou{er} his hede in respect of the triplat than had in hym-self cubikly,[{28}] [[it setteth{e} a-way all{e} his respect]] or as nygh{e} as ye may.

+----------------+--+-+-+-+-+-+---++--+-+-+-+-+--++----+-+--+-+--+ | Residuu{m} | | | | | | | 5 || | | | | | 4|| 1|0|1 |9| | +----------------+--+-+-+-+-+-+---++--+-+-+-+-+--++----+-+--+-+--+ | Cubicandu{s} | 8|3|6|5|4|3| 2 || 3|0|0|7|6| 7|| 1 1|6|6 |7| | +----------------+--+-+-+-+-+-+---++--+-+-+-+-+--++----+-+--+-+--+ | Triplum | | |6|0| | | || | | |1|8| || | |4 | | | +----------------+--+-+-+-+-+-+---++--+-+-+---+--++----+-+--+-+--+ | Subt{r}iplu{m} | 2| | |0| | |[3]|| | |6| | | 7|| 2| | |2| | +----------------+--+-+-+-+-+-+---++--+-+-+-+-+--++----+-+--+-+--+

[Sidenote: Continue this process till the first figure is reached. Examples. The residue. Special cases. Special case.]

Nother me shall{e} not cesse of the fyndyng{e} of that digit, neither of his triplacio{u}n, neþ{er} of the triplat-is [{29}]anteriorac{i}o{u}n, that is to sey, settyng forward{e} by .2. differences, Ne therof the vndre-triple to be put vndre the triple, Nether of the multiplicacio{u}n þ{er}of, Neither of the subtraccio{u}n, till{e} it come to the first figure, vnder the which{e} is a digitall{e} nombre to be found{e}, the which{e} with{e} his vndre-treblis most be had{e} in tribles, After-ward{e} w{i}t{h}out vnder-treblis to be had{e} into produccio{u}n, settyng away all{e} that is ou{er} the hed{e} of the triplat nombre, After had into hymself{e} cubikly, and sette all{e}-way that is ou{er} hym.

+------------------+---+---+---+---++---+---+---+---+---+ | To be cubiced{e} | 1 | 7 | 2 | 8 || 3 | 2 | 7 | 6 | 8 | +------------------+---+---+---+---++---+---+---+---+---+ | The triple | | | 3 | 2 || | | | 9 | | +------------------+---+---+---+---++---+---+---+---+---+ | The vnder triple | | | 1 | 2 || |[3]| | 3 | 3 | +------------------+---+---+---+---++---+---+---+---+---+

Also note wele that the p{ro}ducc{i}on comyng{e} of the ledyng of a digite found{e}[{30}] [[w{i}t{h} an vndre-triple / other of an vndre-triple in a triple or triplat is And after-ward{e} w{i}t{h} out vndre-triple other vndre-triplis in the p{ro}duct and ayene that p{ro}duct that cometh{e} of the ledyng{e} of a digit found{e} in hym-self{e} cubicall{e}]] me may adde to, and also w{i}t{h}-draw fro of the totall{e} nombre sette above that digit so found{e}.[{31}] [[as ther had be a divisio{u}n made as it is opened{e} before]] That done ought or nought most be the residue. If it be nought, It is open that the nombre p{ro}posed{e} was a cubik{e} nombre, And his rote a digit founde last w{i}t{h} the vnder-triples: If the rote therof wex bad{e} in hym-self{e}, and afterward{e} p{ro}duct they shall{e} make the first fig{ur}es. And yf ought be in residue, kepe that w{i}t{h}out in the table; and it is open{e} that the nombre was not a cubik{e}. but a digit last founde w{i}t{h} the vndirtriplis is rote of the most cubik{e} vndre the nombre p{ro}posed{e} conteyned{e}, the which{e} rote yf it be had{e} in hym-self{e}, And aft{er}ward{e} in a p{ro}duct of that shall{e} growe the most cubik{e} vndre the nombre p{ro}posed{e} conteyned{e}, And yf that be added{e} to a cubik{e} the residue res{er}ued{e} in the table, woll{e} make the same figures that ye had{e} first. [*Fol. 56.] And yf no digit after the anterioracio{u}n[{32}] may not be found{e}, than put ther{e} a cifre vndre a cifre vndir the third{e} figure, And put forward{e} þe fig{ur}es. Note also wele that yf in the nombre p{ro}posed{e} ther ben no place of thowsand{es}, me most begynne vnder the first figure in the extraccio{u}n of the rote. some vsen forto distingue the nombre by threes, and ay begynne forto wirch{e} vndre the first of the last t{er}nary other unco{m}plete nombre, the which{e} maner of op{er}acio{u}n accordeth{e} w{i}t{h} that before. And this at this tyme suffiseth{e} in extraccio{u}n of nombres quadrat or cubik{es} {et}c.

[Sidenote: Examples.]

+-------------------+---+--+------+--+--+--+--++--+--+--+--+--+--+--+ | The residue | | | | | | | 0|| | | | | | 1| 1| +-------------------+---+--+------+--+--+--+--++--+--+--+--+--+--+--+ | The cubicand{us} | 8 | 0| 0 | 0| 0| 0| 0|| 8| 2| 4| 2| 4| 1| 9| +-------------------+---+--+------+--+--+--+--++--+--+--+--+--+--+--+ | The triple | | |[{33}]| 0| 0| | || | | 6| | | | | +-------------------+---+--+------+--+--+--+--++--+--+--+--+--+--+--+ | The vndert{r}iple |[2]| | | 0| 0| | || 2| | | 6| 2| | | +-------------------+---+--+------+--+--+--+--++--+--+--+--+--+--+--+

[Headnote: Table of Numbers, &c.]

[Sidenote: A table of numbers; probably from the Abacus.]

1 2 3 4 5 6 one. x. an. hundred{e}/ a thowsand{e}/ x. thowsand{e}/ An hundred{e} 7 thowsand{e}/ A thowsand{e} tymes a thowsand{e}/ x. thousand{e} tymes

a thousand{e}/ An hundred{e} thousand{e} tymes a thousand{e} A

thousand{e} thousand{e} tymes a thousand{e}/ this is the x place

{et}c.

[Ende.]

FOOTNOTES (The Art of Nombryng):

[1: MS. Materiall{e}.] [2: MS. Formall{e}.] [3: ‘the’ in MS.] [4: ‘be’ in MS.] [5: ‘and’ in MS.] [6: ‘is’ in MS.] [7: 6 in MS.] [8: 0 in MS.] [9: 2 in MS.] [10: _sic._] [11: ‘And’ inserted in MS.] [12: ‘4 the’ inserted in MS.] [13: ‘to’ in MS.] [14: ‘that’ repeated in MS.] [15: ‘1’ in MS.] [16: Blank in MS.] [17: ‘nought’ in MS.] [18: 3 written for 2 in MS.] [19: 7 in MS.] [20: runs on in MS.] [21: ‘so’ in MS.] [22: ‘nought’ in MS.] [23: MS. adds here: ‘wher-vpon{e} se the table in the next side of the next leef{e}.’] [24: 110 in MS.] [25: 0 in MS.] [26: double in MS.] [27: ‘it hym-self{e}’ in MS.] [28: MS. adds here: ‘it setteth{e} a-way all{e} his respect.’] [29: ‘aucterioracio{u}n’ in MS.] [30: MS. adds here: ’w{i}t{h} an vndre-triple / other of an vndre-triple in a triple or triplat is And after-ward{e} w{i}t{h} out vndre-triple other vndre-triplis in the p{ro}duct and ayene that p{ro}duct that cometh{e} of the ledyng{e} of a digit found{e} in hym-self{e} cubicall{e}’ /] [31: MS. adds here: ‘as ther had be a divisio{u}n made as it is opened{e} before.’] [32: MS. anteriocacio{u}n.] [33: 4 in MS.]

Accomptynge by counters.

[Transcriber’s Note:

The original text was printed as a single continuous paragraph, with no break between speakers; all examples were shown inline. It has been broken up for this e-text.]

[*116b]

¶ The seconde dialoge of accomptynge by counters.

_Mayster._

Nowe that you haue learned the commen kyndes of Arithmetyke with the penne, you shall se the same art in cou{n}ters: whiche feate doth not only serue for them that can not write and rede, but also for them that can do bothe, but haue not at some tymes theyr penne or tables redye with them. This sorte is in two fourmes co{m}menly. The one by lynes, and the other without lynes: in that y^t hath lynes, the lynes do stande for the order of places: and in y^t that hath no lynes, there must be sette in theyr stede so many counters as shall nede, for eche lyne one, and they shall supplye the stede of the lynes.

_S._ By examples I shuld better p{er}ceaue your meanynge.

_M._ For example of the [*117a.] ly[*]nes:

----1-0-0-0-0-0-- ----1-0-0-0-0---- -X--1-0-0-0------ ----1-0-0-------- ----1-0---------- ----1------------

[Sidenote: Numeration.]

Lo here you se .vi. lynes whiche stande for syxe places so that the nethermost standeth for y^e fyrst place, and the next aboue it, for the second: and so vpward tyll you come to the hyghest, which is the syxte lyne, and standeth for the syxte place. Now what is the valewe of euery place or lyne, you may perceaue by the figures whiche I haue set on them, which is accordynge as you learned before in the Numeration of figures by the penne: for the fyrste place is the place of vnities or ones, and euery counter set in that lyne betokeneth but one: {and} the seconde lyne is the place of 10, for euery counter there, standeth for 10. The thyrd lyne the place of hundredes: the fourth of thousandes: {and} so forth.

_S._ Syr I do perceaue that the same order is here of lynes, as was in the other figures [*117b] by places, so that you shall not nede longer to stande about Numeration, excepte there be any other difference.

_M._ Yf you do vndersta{n}de it, then how wyll you set 1543?

_S._ Thus, as I suppose.

-X--1-- ----5-- ----4-- ----3--

_M._ You haue set y^e places truely, but your figures be not mete for this vse: for the metest figure in this behalfe, is the figure of a cou{n}ter round, as you se here, where I haue expressed that same summe.

-X--o-------- o -------------

----o-o-o-o--

----o-o-o----

_S._ So that you haue not one figure for 2, nor 3, nor 4, and so forth, but as many digettes as you haue, you set in the lowest lyne: and for euery 10 you set one in the second line: and so of other. But I know not by what reason you set that one counter for 500 betwene two lynes.

_M._ you shall remember this, that when so euer you nede to set downe 5, 50, or 500, or 5000, or so forth any other nomber, whose numerator [*118a] is 5, you shall set one counter for it, in the next space aboue the lyne that it hath his denomination of, as in this example of that 500, bycause the numerator is 5, it must be set in a voyd space: and bycause the denominator is hundred, I knowe that his place is the voyde space next aboue hundredes, that is to say, aboue the thyrd lyne. And farther you shall marke, that in all workynge by this sorte, yf you shall sette downe any summe betwene 4 and 10, for the fyrste parte of that nomber you shall set downe 5, & then so many counters more, as there reste no{m}bers aboue 5. And this is true bothe of digettes and articles. And for example I wyll set downe this su{m}me 287965,

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which su{m}me yf you marke well, you nede none other exa{m}ples for to lerne the numeration of [*118b] this forme. But this shal you marke, that as you dyd in the other kynde of arithmetike, set a pricke in the places of thousa{n}des, in this worke you shall sette a starre, as you se here.

[Headnote: Addition on the Counting Board.]

[Sidenote: Addition.]

_S._ Then I perceave numeration, but I praye you, howe shall I do in this arte to adde two summes or more together?

_M._ The easyest way in this arte is, to adde but 2 su{m}mes at ones together: how be it you may adde more, as I wyll tell you anone. Therfore when you wyll adde two su{m}mes, you shall fyrst set downe one of them, it forseth not whiche, {and} then by it drawe a lyne crosse the other lynes. And afterward set downe the other su{m}me, so that that lyne may be betwene them, as yf you wolde adde 2659 to 8342, you must set your su{m}mes as you se

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