Chapter IV: Part 4

And ther ben .2. causes whan the last figure may not be sette vnder the last, other that the last of the lower nombre may not be w{i}t{h}-draw of the last of the ou{er}er nombre for it is lasse than the lower, other how be it, that it myght be w{i}t{h}-draw as for hym-self fro the ou{er}er the remenaunt may not so oft of them above, other yf þe last of the lower be even to the figure above his hede, and þe next last oþ{er} the figure be-fore þ{a}t be more þan the figure above sette. [*Fol. 53^2.] These so ordeyned{e}, me most wirch{e} from the last figure of þe nombre of the dyvyser, and se how oft it may be w{i}t{h}-draw of and fro the figure aboue his hede, namly so that the remen{au}nt may be take of so oft, and to se the residue as here:--

[Sidenote: An example.]

+------------------+---+---+---+
| The residue | | 2 | 6 |
+------------------+---+---+---+
| The quocient | | | 9 |
+------------------+---+---+---+
| To be dyvyded{e} | 3 | 3 | 2 |
+------------------+---+---+---+
| The dyvyser | | 3 | 4 |
+------------------+---+---+---+

[Sidenote: Where to set the quotiente. Examples.]

And note wele that me may not with{e}-draw more than .9. tymes nether lasse than ones. Therfor se how oft þe figures of the lower ordre may be w{i}t{h}-draw fro the figures of the ou{er}er, and the nombre that shew{i}t{h} þe q{u}ocient most be writ ou{er} the hede of þat figure, vnder the which{e} the first figure is, of the dyviser; And by that figure me most with{e}-draw all{e} oþ{er} figures of the lower ordir and that of the figures aboue thair{e} hedis. This so don{e}, me most sette forward{e} þe figures of the diuiser by o difference toward{es} the right hond{e} and worch{e} as before; and thus:--

+--------------+---+---+---+---+---+---++---+---+---+---+---+---+---+
| Residuu{m} | | | | | | || | | | | . | 1 | 2 |
+--------------+---+---+---+---+---+---++---+---+---+---+---+---+---+
| quo{ciens} | | | | 6 | 5 | 4 || | | | 2 | 0 | 0 | 4 |
+--------------+---+---+---+---+---+---++---+---+---+---+---+---+---+
| Diuidend{us} | 3 | 5 | 5 | 1 | 2 | 2 || 8 | 8 | 6 | 3 | 7 | 0 | 4 |
+--------------+---+---+---+---+---+---++---+---+---+---+---+---+---+
| Diuisor | | 5 | 4 | 3 | | || 4 | 4 | 2 | 3 | | | |
+--------------+---+---+---+---+---+---++---+---+---+---+---+---+---+

+------------------+---+---+---+---+---+---+
| The quocient | | | | 6 | 5 | 4 |
+------------------+---+---+---+---+---+---+
| To be dyvyded{e} | 3 | 5 | 5 | 1 | 2 | 2 |
+------------------+---+---+---+---+---+---+
| The dyvyser | | 5 | 4 | 3 | | |
+------------------+---+---+---+---+---+---+

[Sidenote: A special case.]

And yf it happ{e} after þe settyng forward{e} of the fig{ur}es þ{a}t þe last of the divisor may not so oft be w{i}t{h}draw of the fig{ur}e above his hede, above þat fig{ur}e vnder the which{e} the first of the diuiser is writ me most sette a cifre in ordre of the nombre quocient, and sette the fig{ur}es forward{e} as be-fore be o difference alone, and so me shall{e} do in all{e} nombres to be dyvided{e}, for where the dyviser may not be w{i}t{h}-draw me most sette there a cifre, and sette forward{e} the figures; as here:--

+------------------+---+---+---+---+---+---+---+
| The residue | | | | | | 1 | 2 |
|------------------+---+---+---+---+---+---+---+
| The quocient | | | | 2 | 0 | 0 | 4 |
|------------------+---+---+---+---+---+---+---+
| To be dyvyded{e} | 8 | 8 | 6 | 3 | 7 | 0 | 4 |
|------------------+---+---+---+---+---+---+---+
| The dyvyser | 4 | 4 | 2 | 3 | | | |
+------------------+---+---+---+---+---+---+---+

[Sidenote: Another example. What the quotient shows. How to prove
your division, or multiplication.]

And me shall{e} not cesse fro such{e} settyng of fig{ur}es forward{e}, nether of settyng{e} of þe quocient into the dyviser, neþ{er} of subt{ra}ccio{u}n of the dyvyser, till{e} the first of the dyvyser be w{i}t{h}-draw fro þe first to be divided{e}. The which{e} don{e}, or ought,[{17}] oþ{er} nought shall{e} remayne: and yf it be ought,[{17}] kepe it in the tables, And eu{er} vny it to þe diviser. And yf þ{o}u wilt wete how many vnytees of þe divisio{u}n [*Fol. 53^3.] wol growe to the nombre of the diviser{e}, the nombre quocient wol shewe it: and whan such{e} divisio{u}n is made, and þ{o}u lust p{ro}ve yf thow have wele done or no, Multiplie the quocient by the diviser, And the same fig{ur}es wolle come ayene that thow haddest bifore and none other. And yf ought be residue, than w{i}t{h} addicio{u}n therof shall{e} come the same figures: And so multiplicacio{u}n p{ro}vith{e} divisio{u}n, and dyvisio{u}n multiplicacio{u}n: as thus, yf multiplicacio{u}n be made, divide it by the multipliant, and the nombre quocient wol shewe the nombre that was to be multiplied{e}, {et}c.

[Headnote: Chapter VIII. Progression.]

[Sidenote: Definition of Progression. Natural Progression. Broken
Progression. The 1st rule for Natural Progression. The second rule.
The first rule of Broken Progression. The second rule.]

Progressio{u}n is of nombre after egall{e} excesse fro oone or tweyn{e} take ag{r}egacio{u}n. of p{ro}gressio{u}n one is naturell{e} or co{n}tynuell{e}, þ{a}t oþ{er} broken and discontynuell{e}. Naturell{e} it is, whan me begynneth{e} w{i}t{h} one, and kepeth{e} ordure ou{er}lepyng one; as .1. 2. 3. 4. 5. 6., {et}c., so þ{a}t the nombre folowyng{e} passith{e} the other be-fore in one. Broken it is, whan me lepith{e} fro o nombre till{e} another, and kepith{e} not the contynuel ordir{e}; as 1. 3. 5. 7. 9, {et}c. Ay me may begynne w{i}t{h} .2., as þus; .2. 4. 6. 8., {et}c., and the nombre folowyng passeth{e} the others by-fore by .2. And note wele, that naturell{e} p{ro}gressio{u}n ay begynneth{e} w{i}t{h} one, and Int{er}cise or broken p{ro}gressio{u}n, omwhile begynnyth{e} w{i}th one, omwhile w{i}t{h} twayn{e}. Of p{ro}gressio{u}n naturell .2. rules ther be yove, of the which{e} the first is this; whan the p{ro}gressio{u}n naturell{e} endith{e} in even nombre, by the half therof multiplie þe next totall{e} ou{er}er{e} nombre; Example of grace: .1. 2. 3. 4. Multiplie .5. by .2. and so .10. cometh{e} of, that is the totall{e} nombre þ{er}of. The second{e} rule is such{e}, whan the p{ro}gressio{u}n naturell{e} endith{e} in nombre od{e}. Take the more porcio{u}n of the oddes, and multiplie therby the totall{e} nombre. Example of grace 1. 2. 3. 4. 5., multiplie .5. by .3, and thryes .5. shall{e} be resultant. so the nombre totall{e} is .15. Of p{ro}gresio{u}n int{er}cise, ther ben also .2.[{18}] rules; and þe first is þis: Whan the Int{er}cise p{ro}gression endith{e} in even nombre by half therof multiplie the next nombre to þat half{e} as .2.[{18}] 4. 6. Multiplie .4. by .3. so þat is thryes .4., and .12. the nombre of all{e} the p{ro}gressio{u}n, woll{e} folow. The second{e} rule is this: whan the p{ro}gressio{u}n int{er}scise endith{e} in od{e}, take þe more porcio{u}n of all{e} þe nombre, [*Fol. 53^4.] and multiplie by hym-self{e}; as .1. 3. 5. Multiplie .3. by hym-self{e}, and þe some of all{e} wolle be .9., {et}c.

[Headnote: Chapter IX. Extraction of Roots.]

[Sidenote: The preamble of the extraction of roots. Linear,
superficial, and solid numbers. Superficial numbers. Square numbers.
The root of a square number. Notes of some examples of square roots
here interpolated. Solid numbers. Three dimensions of solids. Cubic
numbers. All cubics are solid numbers. No number may be both linear
and solid. Unity is not a number.]

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,

-X-----------

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

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

-------------|-----------
o |
-X--o-o-o----|--o-o------
| o
----o-o-o----|--o--------
| o
----o-o-o-o--|-----------
| o
----o-o------|--o-o-o-o--

here. And then yf you lyst, you [*119a] may adde the one to the other in the same place, or els you may adde them both together in a newe place: which waye, bycause it is moste playnest, I wyll showe you fyrst. Therfore wyl I begynne at the vnites, whiche in the fyrst su{m}me is but 2, {and} in y^e second su{m}me 9, that maketh 11, those do I take vp, and for them I set 11 in the new roume, thus,

-------------|-------|-------
o | |
-X--o-o-o----|--o-o--|-------
| o |
----o-o-o----|--o----|-------
| o |
----o-o-o-o--|-------|-o-----
| |
-------------|-------|-o-----

Then do I take vp all y^e articles vnder a hundred, which in the fyrst su{m}me are 40, and in the second summe 50, that maketh 90: or you may saye better, that in the fyrste summe there are 4 articles of 10, and in the seconde summe 5, which make 9, but then take hede that you sette them in theyr [*119b] ryght lynes as you se here.

-----------|----------|-------------
o | |
-X--o-o-o--|--o-o-----|-------------
| o |
----o-o-o--|--o-------|-------------
| | o
-----------|----------|--o-o-o-o-o--
| |
-----------|----------|--o----------

Where I haue taken awaye 40 fro{m} the fyrste su{m}me, and 50 from y^e second, and in theyr stede I haue set 90 in the thyrde, whiche I haue set playnely y^t you myght well perceaue it: how be it seynge that 90 with the 10 that was in y^e thyrd roume all redy, doth make 100, I myghte better for those 6 cou{n}ters set 1 in the thyrde lyne, thus:

----------

-X--------

----o-----

----------

----o-----

For it is all one summe as you may se, but it is beste, neuer to set 5 cou{n}ters in any line, for that may be done with 1 cou{n}ter in a hygher place.

_S._ I iudge that good reaso{n}, for many are vnnedefull, where one wyll serue.

_M._ Well, then [*120a] wyll I adde forth of hundredes: I fynde 3 in the fyrste summe, and 6 in the seconde, whiche make 900, them do I take vp {and} set in the thyrd roume where is one hundred all redy, to whiche I put 900, and it wyll be 1000, therfore I set one cou{n}ter in the fourth lyne for them all, as you se here.

-----------|-------|--------
o | |
-X--o-o-o--|--o-o--|--o-----
| |
-----------|-------|--------
| |
-----------|-------|--------
| |
-----------|-------|--o-----

Then adde I y^e thousandes together, whiche in the fyrst su{m}me are 8000, {and} in y^e second 2000, that maketh 10000: them do I take vp fro{m} those two places, and for them I set one counter in the fyfte lyne, and then appereth as you se, to be 11001, for so many doth amount of the addition of 8342 to 2659.

----o-----

-X--o-----

----------

----------

----o-----

[*120b] _S._ Syr, this I do perceave: but how shall I set one su{m}me to an other, not chaungynge them to a thyrde place?

_M._ Marke well how I do it: I wyll adde together 65436, and 3245, whiche fyrste I set downe thus.

-------------|--------------
| o
-------------|--o-----------
| o
-X--o-o-o----|--------------
|
----o-o------|--o-o-o-o-----
|
----o-o-o-o--|--o-o-o-------
o | o
-------------|--o-----------

Then do I begynne with the smalest, which in the fyrst summe is 5, that do I take vp, and wold put to the other 5 in the seconde summe, sauynge that two counters can not be set in a voyd place of 5, but for them bothe I must set 1 in the seconde lyne, which is the place of 10, therfore I take vp the 5 of the fyrst su{m}me, {and} the 5 of the seco{n}de, and for them I set 1 in the seco{n}d lyne, [*121a] as you se here.

-------------|--------------
| o
-------------|--o-----------
| o
-X--o-o-o----|--------------
|
----o-o------|--o-o-o-o-----
|
----o-o-o-o--|--o-o-o-o-----
|
-------------|--o-----------

Then do I lyke wayes take vp the 4 counters of the fyrste su{m}me {and} seconde lyne (which make 40) and adde them to the 4 counters of the same lyne, in the second su{m}me, and it maketh 80, But as I sayde I maye not conueniently set aboue 4 cou{n}ters in one lyne, therfore to those 4 that I toke vp in the fyrst su{m}me, I take one also of the seconde su{m}me, and then haue I taken vp 50, for whiche 5 counters I sette downe one in the space ouer y^e second lyne, as here doth appere.

-----------|--------------
| o
-----------|--o-----------
| o
-X--o-o-o--|--------------
|
----o-o----|--o-o-o-o-----
| o
-----------|--o-o-o-------
|
-----------|--o-----------

[*121b.] and then is there 80, as well w^t those 4 counters, as yf I had set downe y^e other 4 also. Now do I take the 200 in the fyrste su{m}me, and adde them to the 400 in the seconde summe, and it maketh 600, therfore I take vp the 2 counters in the fyrste summe, and 3 of them in the seconde summe, and for them 5 I set 1 in y^e space aboue, thus.

-----------|------------
| o
-----------|--o---------
| o
-X--o-o-o--|------------
| o
-----------|--o---------
| o
-----------|--o-o-o-----
|
-----------|--o---------

Then I take y^e 3000 in y^e fyrste su{m}me, vnto whiche there are none in the second summe agreynge, therfore I do onely remoue those 3 counters from the fyrste summe into the seconde, as here doth appere.

----|-------------
| o
----|---o---------
| o
-X--|---o-o-o-----
| o
----|-o-----------
| o
----|---o-o-o-----
|
----|---o---------

[*122a.] And so you see the hole su{m}me, that amou{n}teth of the addytio{n} of 65436 with 3245 to be 6868[1]. And yf you haue marked these two exa{m}ples well, you nede no farther enstructio{n} in Addition of 2 only summes: but yf you haue more then two summes to adde, you may adde them thus. Fyrst adde two of them, and then adde the thyrde, and y^e fourth, or more yf there be so many: as yf I wolde adde 2679 with 4286 and 1391. Fyrste I adde the two fyrste summes thus.

-------------|-----------|--------------
| | o
-X--o-o------|--o-o-o-o--|--o-----------
o | | o
----o--------|--o-o------|--o-o-o-o-----
o | o | o
----o-o------|--o-o-o----|--o-----------
o | o | o
----o-o-o-o--|--o--------|--------------

[*122b.] And then I adde the thyrde thereto thus. And so of more yf you haue them.

-------------|-----------|------------
| o | o
-X--o--------|--o--------|--o-o-o-----
| o |
----o-o-o----|--o-o-o-o--|--o-o-o-----
o | o | o
----o-o-o-o--|----o------|------------
| o | o
----o--------|-----------|--o---------

[Headnote: Subtraction on the Counting Board.]

[Sidenote: Subtraction.]

_S._ Nowe I thynke beste that you passe forth to Subtraction, except there be any wayes to examyn this maner of Addition, then I thynke that were good to be knowen nexte.

_M._ There is the same profe here that is in the other Addition by the penne, I meane Subtraction, for that onely is a sure waye: but consyderynge that Subtraction must be fyrste knowen, I wyl fyrste teache you the arte of Subtraction, and that by this example: I wolde subtracte 2892 out of 8746. These summes must I set downe as I dyd in Addition: but here it is best [*116a (_sic_).] to set the lesser no{m}ber fyrste, thus.

-------------|--------------
| o
-X--o-o------|--o-o-o-------
o | o
----o-o-o----|--o-o---------
o |
----o-o-o-o--|--o-o-o-o-----
| o
----o-o------|--o-----------

Then shall I begynne to subtracte the greatest nombres fyrste (contrary to the vse of the penne) y^t is the thousandes in this exa{m}ple: therfore I fynd amongest the thousandes 2, for which I withdrawe so many fro{m} the seconde summe (where are 8) and so remayneth there 6, as this exa{m}ple showeth.

-------------+--------------
| o
-+-----------+--o-----------
o | o
----o-o-o----+--o-o---------
o |
----o-o-o-o--+--o-o-o-o-----
| o
----o-o------+--o-----------

Then do I lyke wayes with the hundredes, of whiche in the fyrste summe [*116b] I fynde 8, and is the seconde summe but 7, out of whiche I can not take 8, therfore thus muste I do: I muste loke how moche my summe dyffereth from 10, whiche I fynde here to be 2, then must I bate for my su{m}me of 800, one thousande, and set downe the excesse of hundredes, that is to saye 2, for so moche 100[0] is more then I shuld take vp. Therfore fro{m} the fyrste su{m}me I take that 800, and from the second su{m}me where are 6000, I take vp one thousande, and leue 5000; but then set I downe the 200 unto the 700 y^t are there all redye, and make them 900 thus.

-------------+--------------
| o
-+-----------+--------------
| o
-------------+--o-o-o-o-----
o |
----o-o-o-o--+--o-o-o-o-----
| o
----o-o------+--o-----------

Then come I to the articles of te{n}nes where in the fyrste su{m}me I fynde 90, [*117a] and in the seconde su{m}me but only 40: Now consyderyng that 90 can not be bated from 40, I loke how moche y^t 90 doth dyffer from the next summe aboue it, that is 100 (or elles whiche is all to one effecte, I loke how moch 9 doth dyffer fro{m} 10) {and} I fynd it to be 1, then in the stede of that 90, I do take from the second summe 100: but consyderynge that it is 10 to moche, I set downe 1 in y^e nexte lyne beneth for it, as you se here.

---------+------------
| o
-+-------+------------
| o
---------+--o-o-o-----
| o
---------+------------
| o
----o-o--+--o---------

Sauynge that here I haue set one counter in y^e space in stede of 5 in y^e nexte lyne. And thus haue I subtracted all saue two, which I must bate from the 6 in the second summe, and there wyll remayne 4, thus.

----+--------------
| o
-+--+--------------
| o
----+--o-o-o-------
| o
----+--------------
|
----+--o-o-o-o-----

So y^t yf I subtracte 2892 fro{m} 8746, the remayner wyll be 5854, [*117b] And that this is truely wrought, you maye proue by Addition: for yf you adde to this remayner the same su{m}me that you dyd subtracte, then wyll the formar su{m}me 8746 amount agayne.

_S._ That wyll I proue: and fyrst I set the su{m}me that was subtracted, which was 2892, {and} the{n} the remayner 5854, thus.

--------------+--------------
| o
-||--o-o------+--------------
o | o
-----o-o-o----+--o-o-o-------
o | o
-----o-o-o-o--+--------------
|
-----o-o------+--o-o-o-o-----

Then do I adde fyrst y^e 2 to 4, whiche maketh 6, so take I vp 5 of those counters, and in theyr stede I sette 1 in the space, as here appereth.

--------------+------------
| o
-||--o-o------+------------
o | o
-----o-o-o----+--o-o-o-----
o | o
-----o-o-o-o--+------------
| o
--------------+--o---------

[*118a] Then do I adde the 90 nexte aboue to the 50, and it maketh 140, therfore I take vp those 6 counters, and for them I sette 1 to the hundredes in y^e thyrde lyne, {and} 4 in y^e second lyne, thus.

------------+--------------
| o
-||--o-o----+--------------
o | o
-----o-o-o--+--o-o-o-o-----
|
------------+--o-o-o-o-----
| o
------------+----o---------

Then do I come to the hundredes, of whiche I fynde 8 in the fyrst summe, and 9 in y^e second, that maketh 1700, therfore I take vp those 9 counters, and in theyr stede I sette 1 in the .iiii. lyne, and 1 in the space nexte beneth, and 2 in the thyrde lyne, as you se here.

----------+--------------
| o
-||--o-o--+--o-----------
| o
----------+--o-o---------
|
----------+--o-o-o-o-----
| o
----------+--o-----------

Then is there lefte in the fyrste summe but only 2000, whiche I shall take vp from thence, and set [*118b] in the same lyne in y^e second su{m}me, to y^e one y^t is there all redy: {and} then wyll the hole su{m}me appere (as you may wel se) to be 8746, which was y^e fyrst grosse summe, {and} therfore I do perceaue, that I hadde well subtracted before. And thus you may se how Subtraction maye be tryed by Addition.

----+--------------
| o
-X--+--o-o-o-------
| o
----+--o-o---------
|
----+--o-o-o-o-----
| o
----+----o---------

_S._ I perceaue the same order here w^t cou{n}ters, y^t I lerned before in figures.

_M._ Then let me se howe can you trye Addition by Subtraction.

_S._ Fyrste I wyl set forth this exa{m}ple of Additio{n} where I haue added 2189 to 4988, and the hole su{m}me appereth to be 7177,

--------------+-----------+----------
| | o
-||--o-o------+--o-o-o-o--+--o-o-----
| o |
-----o--------+--o-o-o-o--+--o-------
o | o | o
-----o-o-o----+--o-o-o----+--o-o-----
o | o | o
-----o-o-o-o--+--o-o-o----+--o-o-----

[*119a] Nowe to trye whether that su{m}me be well added or no, I wyll subtract one of the fyrst two su{m}mes from the thyrd, and yf I haue well done y^e remayner wyll be lyke that other su{m}me. As for example: I wyll subtracte the fyrste summe from the thyrde, whiche I set thus in theyr order.

--------------+----------
| o
-||--o-o------+--o-o-----
|
-----o--------+--o-------
o | o
-----o-o-o----+--o-o-----
o | o
-----o-o-o-o--+--o-o-----

Then do I subtract 2000 of the fyrste summe fro{m} y^e second su{m}me, and then remayneth there 5000 thus.

-------------+----------
| o
-X-----------+-----------
|
----o--------+--o-------
o | o
----o-o-o----+--o-o-----
o | o
----o-o-o-o--+--o-o-----

Then in the thyrd lyne, I subtract y^e 100 of the fyrste summe, fro{m} the second su{m}me, where is onely 100 also, and then in y^e thyrde lyne resteth nothyng. Then in the second lyne with his space ouer hym, I fynde 80, which I shuld subtract [*119b] from the other su{m}me, then seyng there are but only 70 I must take it out of some hygher summe, which is here only 5000, therfore I take vp 5000, and seyng that it is to moch by 4920, I sette downe so many in the seconde roume, whiche with the 70 beynge there all redy do make 4990, & then the summes doth stande thus.

--------------+--------------
|
-||-----------+--o-o-o-o-----
| o
--------------+--o-o-o-o-----
| o
--------------+--o-o-o-o-----
o | o
-----o-o-o-o--+--o-o---------

Yet remayneth there in the fyrst su{m}me 9, to be bated from the second summe, where in that place of vnities dothe appere only 7, then I muste bate a hygher su{m}me, that is to saye 10, but seynge that 10 is more then 9 (which I shulde abate) by 1, therfore shall I take vp one counter from the seconde lyne, {and} set downe the same in the fyrst [*120a] or lowest lyne, as you se here.

-----+--------------
|
-||--+--o-o-o-o-----
| o
-----+--o-o-o-o-----
| o
-----+--o-o-o-------
| o
-----+--o-o-o-------

And so haue I ended this worke, {and} the su{m}me appereth to be y^e same, whiche was y^e seconde summe of my addition, and therfore I perceaue, I haue wel done.

_M._ To stande longer about this, it is but folye: excepte that this you maye also vnderstande, that many do begynne to subtracte with counters, not at the hyghest su{m}me, as I haue taught you, but at the nethermoste, as they do vse to adde: and when the summe to be abatyd, in any lyne appeareth greater then the other, then do they borowe one of the next hygher roume, as for example: yf they shuld abate 1846 from 2378, they set y^e summes thus.

--------------+------------
|
-||--o--------+--o-o-------
o |
-----o-o-o----+--o-o-o-----
| o
-----o-o-o-o--+--o-o-------
o | o
-----o--------+--o-o-o-----

[*120b] And fyrste they take 6 whiche is in the lower lyne, and his space from 8 in the same roumes, in y^e second su{m}me, and yet there remayneth 2 counters in the lowest lyne. Then in the second lyne must 4 be subtracte from 7, and so remayneth there 3. Then 8 in the thyrde lyne and his space, from 3 of the second summe can not be, therfore do they bate it from a hygher roume, that is, from 1000, and bycause that 1000 is to moch by 200, therfore must I sette downe 200 in the thyrde lyne, after I haue taken vp 1000 from the fourth lyne: then is there yet 1000 in the fourth lyne of the fyrst summe, whiche yf I withdrawe from the seconde summe, then doth all y^e figures stande in this order.

-----+------------
|
-||--+------------
| o
-----+------------
|
-----+--o-o-o-----
|
-----+--o-o-------

So that (as you se) it differeth not greatly whether you begynne subtractio{n} at the hygher lynes, or at [*121a] the lower. How be it, as some menne lyke the one waye beste, so some lyke the other: therfore you now knowyng bothe, may vse whiche you lyst.

[Headnote: Multiplication by Counters.]

[Sidenote: Multiplication.]

But nowe touchynge Multiplicatio{n}: you shall set your no{m}bers in two roumes, as you dyd in those two other kyndes, but so that the multiplier be set in the fyrste roume. Then shall you begyn with the hyghest no{m}bers of y^e seconde roume, and multiply them fyrst after this sort. Take that ouermost lyne in your fyrst workynge, as yf it were the lowest lyne, setting on it some mouable marke, as you lyste, and loke how many counters be in hym, take them vp, and for them set downe the hole multyplyer, so many tymes as you toke vp counters, reckenyng, I saye that lyne for the vnites: {and} when you haue so done with the hygheest no{m}ber then come to the nexte lyne beneth, {and} do euen so with it, and so with y^e next, tyll you haue done all. And yf there be any nomber in a space, then for it [*121b] shall you take y^e multiplyer 5 tymes, and then must you recken that lyne for the vnites whiche is nexte beneth that space: or els after a shorter way, you shall take only halfe the multyplyer, but then shall you take the lyne nexte aboue that space, for the lyne of vnites: but in suche workynge, yf chau{n}ce your multyplyer be an odde nomber, so that you can not take the halfe of it iustly, then muste you take the greater halfe, and set downe that, as if that it were the iuste halfe, and farther you shall set one cou{n}ter in the space beneth that line, which you recken for the lyne of vnities, or els only remoue forward the same that is to be multyplyed.

_S._ Yf you set forth an example hereto I thynke I shal perceaue you.

_M._ Take this exa{m}ple: I wold multiply 1542 by 365, therfore I set y^e nombers thus.

------------+--------------
|
-||---------+--o-----------
| o
-----o-o-o--+--------------
o |
-----o------+--o-o-o-o-----
o |
------------+--o-o---------

[*122a] Then fyrste I begynne at the 1000 in y^e hyghest roume, as yf it were y^e fyrst place, & I take it vp, settynge downe for it so often (that is ones) the multyplyer, which is 365, thus, as you se here:

-----------+-----------+------------
| |
-----------+-----------+--o-o-o-----
| | o
-----------+-----------+--o---------
| | o
-X---------+-----------+------------ [<-]
| o |
----o-o-o--+-----------+------------
o | |
----o------+--o-o-o-o--+------------
o | |
-----------+--o-o------+------------

where for the one counter taken vp from the fourth lyne, I haue sette downe other 6, whiche make y^e su{m}me of the multyplyer, reckenynge that fourth lyne, as yf it were the fyrste: whiche thyng I haue marked by the hand set at the begynnyng of y^e same,

_S._ I perceaue this well: for in dede, this summe that you haue set downe is 365000, for so moche doth amount [*122b] of 1000, multiplyed by 365.

_M._ Well the{n} to go forth, in the nexte space I fynde one counter which I remoue forward but take not vp, but do (as in such case I must) set downe the greater halfe of my multiplier (seyng it is an odde no{m}ber) which is 182, {and} here I do styll let that fourth place stand, as yf it were y^e fyrst:

------------+-----------+--o-o-o--+--o---------
| | o | o
------------+-----------+--o------+--o-o-o-----
| | o |
-||---------+-----------+---------+--o-o------- [<-]
| | | o
-----o-o-o--+-----------+---------+------------
o | | |
-----o------+--o-o-o-o--+---------+------------
o | | |
------------+--o-o------+---------+------------

as in this fourme you se, where I haue set this multiplycatio{n} with y^e other: but for the ease of your vndersta{n}dynge, I haue set a lytell lyne betwene them: now shulde they both in one su{m}me stand thus.

------------+-----------+--o-o-o-o-o-----
| |
------------+-----------+--o-o-o-o-------
| | o
-||---------+-----------+--o-o----------- [<-]
| | o
-----o-o-o--+-----------+----------------
o | |
-----o------+--o-o-o-o--+----------------
o | |
------------+--o-o------+----------------

[*123a] Howe be it an other fourme to multyplye suche cou{n}ters i{n} space is this: Fyrst to remoue the fynger to the lyne nexte benethe y^e space, {and} then to take vp y^e cou{n}ter, {and} to set downe y^e multiplyer .v. tymes, as here you se.

---------+---------+-o-o-o-+------+------+------+------+------+-
| | o | | | | | |
---------+---------+-o-----+o-o-o-+o-o-o-+o-o-o-+o-o-o-+o-o-o-+-
| | o | o | o | o | o | o |
---------+---------+-------+o-----+------+o-----+o-----+o-----+-
| | | o | o | o | o | o |
[->]-X-o-o-o-+---------+-------+------+------+------+------+------+-
o | | | | | | | |
---o-----+-o-o-o-o-+-------+------+------+------+------+------+-
o | | | | | | | |
---------+-o-o-----+-------+------+------+------+------+------+-

Which su{m}mes yf you do adde together into one su{m}me, you shal p{er}ceaue that it wyll be y^e same y^t appeareth of y^e other worki{n}g before, so that [*123b] bothe sortes are to one entent, but as the other is much shorter, so this is playner to reason, for suche as haue had small exercyse in this arte. Not withstandynge you maye adde them in your mynde before you sette them downe, as in this exa{m}ple, you myghte haue sayde 5 tymes 300 is 1500, {and} 5 tymes 60 is 300, also 5 tymes 5 is 25, whiche all put together do make 1825, which you maye at one tyme set downe yf you lyste. But nowe to go forth, I must remoue the hand to the nexte counters, whiche are in the second lyne, and there must I take vp those 4 counters, settynge downe for them my multiplyer 4 tymes, whiche thynge other I maye do at 4 tymes seuerally, or elles I may gather that hole summe in my mynde fyrste, and then set it downe: as to saye 4 tymes 300 is 1200: 4 tymes 60 are 240: and 4 tymes 5 make 20: y^t is in all 1460, y^t shall I set downe also: as here you se. o -----------+-------+-----------+-------------- | | | -----------+-------+--o-o-o-o--+--o----------- | | o | -X---------+-------+--o-o------+--o-o-o-o----- | | o | o ----o-o-o--+-------+-----------+--o----------- o | | | [->] ----o------+-------+-----------+-------------- o | | | -----------+--o-o--+-----------+--------------

[*124a] whiche yf I ioyne in one summe with the formar nombers, it wyll
appeare thus.
o
---------+-------+----------
| | o
---------+-------+--o-------
| |
---------+-------+--o-o-----
| |
--o-o-o--+-------+-o--------
o | |
[->] --o------+-------+----------
o | |
---------+--o-o--+----------

Then to ende this multiplycation, I remoue the fynger to the lowest
lyne, where are onely 2, them do I take vp, and in theyr stede do I set
downe twyse 365, that is 730, for which I set [*124b] one in the space
aboue the thyrd lyne for 500, and 2 more in the thyrd lyne with that one
that is there all redye, and the reste in theyr order, {and} so haue I
ended the hole summe thus.
o
---------+-----+------------
| | o
---------+-----+--o---------
| |
---------+-----+--o-o-------
| | o
--o-o-o--+-----+--o-o-o-----
o | |
--o------+-----+--o-o-o-----
o | |
---------+-----+------------

Wherby you se, that 1542 (which is the nomber of yeares syth Ch[r]ystes incarnation) beyng multyplyed by 365 (which is the nomber of dayes in one yeare) dothe amounte vnto 562830, which declareth y^e no{m}ber of daies sith Chrystes incarnatio{n} vnto the ende of 1542[{1}] yeares. (besyde 385 dayes and 12 houres for lepe yeares).