¶The fyrste parte is of Numeration.

NVmeration is a maner of expressynge of num­bers by certayn figures which are called figurs of Algorisme, the which be tenne, as in this ex­ample.

Of the which nyne be significatyue, the tenth called a siphre, sygnifieng nothyng of it selfe, but only set before the other significatyue fygurs augmentyth theyr signification. In numeration by this crafte ye must euermore begynne at the ryghte syde of the boke, and so towardes the lyft syde, as in this example.

This fygure 9 vnder a, standeth in the fyrste place. 8 vnder b, standeth in the se­conde place, and so forthe to the ende, so [Page] that 3 vnder k, standeth in the laste place. By these tenne figures all maner of number possyble to be excogitate, may cleare­ly and playnely be expressed: whiche all be it, that of them selfe they sygnifie but symple and lyttell number, as ye se afore, yet according to the diuersite of the place they stande in, dyuersly doth theyr signi­fication amount. Wherfore in numeratiō ye muste note twoo thynges, the fygure significatiue, and the place it standeth in for the signification of the fygure depen­deth vpō the number of the place it stan­deth in: For example, this fygure 8 stan­dynge alone, or in the fyrste place signi­fyeth but .viii. but yf he stande in the se­conde place, as here 80 he signifyeth .viii. tymes tenne, whiche is called .iiii. score. yf he stand in the thyrde place, as here 800 he signifyeth .viii. hundreth. &c. Therfore ye must know perfectely the signifycatiō of euery place, before ye can perfectely nū ber. wherfore vnderstand ye, that the first place is a place of vnitees, so that a figur standynge in it, signifieth no more then [Page] though he stand alone. The second place is a place of tennes. The thyrd is a place of hūdrydes. The fourth place is a place of thousandes. The fifth place, a place of tenne thousandes. The .vi. place a place of hundreth thousandes. The .vii. place, is of thousand thousandes, which is cal­led a myllion. The .viii. place, is of tenne myllions. The .ix. is a place of hundreth myllyons. The .x. of thousand myllions. The .xi. of ten thousande myllions. The xii. of a hundreth thousande myllions. The .xiii. of a thousand thousand mylliō, whiche is called myllyon vpon myllyon. And so forth infinitely, euery place ensu­enge, signifieth .x. tymes as moch as the place goynge before. This muste thou knowe perfitely what euery place gyueth and signifyeth: for the place gyueth denomination, and the figure standyng in the same place expresseth how many of the sa­me denomination is to be vnderstand: as in example ye shal more playnly perceyue

In this summe 3400872619 this figure [...] standyth in the .iiii. place, nowe by your [Page] rule afore, the .iiii. place is a place of thousandes, then this fygure 2 standynge in the same place gyueth vs to wytte, that it is two thousande, Lykewyse this fygure 8 standeth in the .vi. place, nowe by your rule afore spoken of, the .vi. place is of hunderth thousandes: then this figure 8 situat in the same place receyueth denominatiō of the place, and representeth to vs viii. hundreth thousandes. Lyke wyse this fygure 1 standeth in the second place and forbycause the second place is a place of tennes, therfore this figure 1 standyng there is bounde to the signification of the place, and so signifyeth one tenne: yf a fy­gure of 4 stode there, it shuld signify .iiii. tennes, that is fourty, and so forth. Then for a farther declaracyon of the foresaide summe, and all other lyke summes. This figure 9 standynge in the fyrste place, sig­nifyeth but hym selfe, that is .ix. This fy­gure 1 standynge in the seconde place, by cause the seconde place is euer a place of tennes, sygnyfyeth one tenne. The fy­gure 6, standynge in the thyrde place, [Page] bycause the thyrde place is a place of hundrethe, doth signifye .vi. hundreth: the fy­gure 2 in the fourth place, signifieth .ii. thousande: the figure 7 forbycause it standethe in the fyfte place, and that place is a place of tenne thousandes, it signifieth vii. tymes tenne thousande, the which is iii. score thousande and ten: the figure 8 in the .vi. place fignifieth .viii. hundreth thousand: the sypher 0 that standeth in y^e vii. place signifieth nothynge, but onely maketh vp a place that the fygures significatyue folowynge maye encrease there signification. Lyke iudgement is of the sypher standynge in the .viii. place: in the ix. place standeth the figure of 4, and this place is a place of hundreth myllyons: therfore this fygure 4 there signifie the iiii, C. myllions. In the .x. place standeth the fygure 3, and this place is a place of thousande myllions: therfore it signify­eth .iii. thousande myllions. So the hole summe is, thre thousand mylions .iiii.C. milliōs .viii.C. thousand, thre score thousandes .xii. thousand .vi. hūdreth, and .xix. [Page] Now to exercyse your selfe in numeratiō numbre with your selfe these summes fo­lowynge, & you shalbe perfecte ynough. Milliō. Mil. Mil. Mil.

Furthermore thou muste note that there be in algorisme thre maner of numbres, Diget number, Article, and Composte.

The digette number, is all maner of numbres, whiche are vnder .x. as these.

The article number is, all numbers whiche are of .x. as these.

The compounde number is all maner of numbres which are compoūd or made [Page] of the dyget & article togyther, as folow.

And so forth of al other. This is sufficiēt for the knowledge of nūbre in Algorisme.

¶The seconde parte called Addition.

ADdition is a collectiō of diuers and sundrye sūmes, into one totall summe, whiche contaynyth as much in hym as al the other sūmes, beynge before sundry. In addition are two nombres to be consyderyd, the one is, the nūbres which muste be adioyned togyther: the other is the nū bers which redoundeth of theyr addition together, whiche otherwyse is called the totall summe. Then when ye wyll adde many summes togyther, fyrst write them fayre the one dyrectely vnder the other, so that the fyrst fygure of the one, be ryght vnder the fyrst of the other, & the seconde [Page] vnder the seconde: euery place correspon­dent vnder other: that done drawe a lyne vnder al these seueral summes, as is to se in y^e exāple folowyng. And whan ye wyll adde your nūbers togyther, begyn at the fyrst places of your sūmes & adde all the figures that ye se in the fyrst places of all your sūmes togither, and y^t that cōmeth of that addition, se whether it be digette number, article, or cōpost: yf it be but dy­get, set y^e dyget benethe the lyne, directely vnder the same first places: yf it be article put a cypher beneth the lyne, right vnder the same fyrst place, & reserue y^e article to be added to the nexte places of thy sūmes & there do likewyse, yf it be cōpost, set the dyget vnder the lyne right vnder y^e same place, & reserue the article in your mynde lykewise to be added to the nexte places of thy sūmes: when the figures standyng in the last places of your summes be ad­ioyned togyther, yf any article or arti­cles remayne, set them downe nexte to the fygure ye sette last before vnder the same lyne: as by examples shall appere.

Your fygures set after this sorte, adde all the fygures that ye fynde in the fyrste places of all the summes together, be­gynnynge at the nethermost sayenge, 2 & 5 is 7, and 4 that is 11, and 6 that is 17, and 7 that is 24, ans 4 that ts 28. This is the hole summe of the figurs added togyther founde in the fyrst places, the whiche nū ­ber is composte: wherfore, as is in your rule, ye must set the dyget ryghte vnder the same place, benethe the lyne, the whi­che is 8, & kepe the articles in your mynd, which is 2. Now to the seconde place, to­warde the lyfte hande, say, 2 that I haue in mynde and 2 is 4, and 2 maketh 6, and 3 is 9, and 5 is 14, and 6 is 20, and 9 is 29, now set the 9 vnder 2, & kepe 2 in mynde, and adde them to the fyrste fygure of the [Page] thyrde place, that is 3. Now say 2 and 3 is 5, and 4 is 9, and 9 is 18, and 4 is 22, and 5 is 27, and 8 is 35. Now set 5 vnder 3, and kepe [...] in mynde. Now to the fourthe place, to­warde the lyfte hande where 4 standeth, now 3 that ye haue in mynde and 4 is 7, & 7 is 14, and 8 is 22, and 3 is 25, & 4 is 29, and 7 is 36, set 6 vnder 4, and kepe 3, and adde that 3 to the vndermost fygure of the syxt summe that is 3, and say 3 and 3 is 6, and 6 is 12, and 7 is 19, and 2 is 21, and 3 is 24, and 6 is 30. All the figures of this place added togyther as ye se, maketh article number wherfore accordyng to your rule set a cy­fer 0 vnder that place benethe the lyne, & the artycle whiche is 3 nexte to the same cyfer, & al is finished. And all these sūmes thus collected togither maketh 306598.

An other example of addition.

Begyn fyrst as ye dyd before, at the fyrst places, addyng them all togyther, begynnynge at the nethermost, sayenge, 1 and 2 is 3, and 1 is 4, and 5 is 9, this is the hole summe of the fygures standynge in the fyrst place, the whiche is diget number, and therfore accordynge to the rule, set it ryghte vnder the same place benethe the lyne: then procede to the second place and begyn at the nether ende sayenge, 2 and 6 is 8, and 5 is 13, and 4 is 17, these number is compost nūber, therfore set y^e diget righte vnder that place beneth the lyne, which is 7, reseruynge the artycle in your mynde, and so to the thyrde place, sayeng, 1 that I haue in my mynde and 4 that is 5, & 2 is 7, and 1 is 8, and 9 is 17, and 8 is [...]5, this number also is cōpost, wherfo [...]e set the diget 5 vnder that thyrde place, & reserue the article 2 in mynde to the nexte place, then to y^e next places sayeng, 2 y^t I haue in mynde & 6 is 8, & 1 is 9, & 5 is 14, & 1 is 15, & 7 is 22, this is also cōpost, therfore set y^e diget 2 vnder that .iiii. place, & reserue the article 2 to y^e next places, then to the fifth place sayeng [Page] 2 that I haue in mynde & 4 is 6, & 3 is 9, & [...] is 15, this is also compost nūber, set the diget 5 vnder y^e fyfth place, and kepe y^e article in mynde, to the .vi. place sayeng, 1 y^e I haue in mynde & 8 is 9 & 5 is 14, and 6 is 20 this is article nūber, therfore accordynge to the rule set a syfer vnder the place be­nethe the lyne, & kepe the article in mynde and cum to the .vii. place, in y^e whiche places for bycause thou fyndest nothyng but cyphers to the whith y^u myghtes adioyne thy article reserued, the which was 2, therfore vnder y^e same .vii. place set that same reserued 2, and then com to the .viii. place and there fyndest y^u nothyng but cyphers wherfore vnder the same place set beneth the lyne a cypher, accordyng to the rule, then c [...]e to the .ix. places & say 5 and 6 is 11, & 1 is 12, the which is cōpost nūber, ther­fore set y^e diget which is 2 vnder the line & reserue y^e article in mynd, which is 1, now for bycause there is no mo places wher­vnto ye myght adde this reserued article therfore according to your rule ye shal set it downe next vnto the figure that ye dyd [Page] set vnder the lyne laste, as is in your ex­ample. This .ii. examples were sufficient ynough to the redynesse of addition, how be it yet that it may be the playner I wyl subscribe an other example.

Adde the fyrst place togyther. Fyrste there thou fyndest nothyng but cyfers, wherfore sette a cyfer vnder the lyne, and so lykewyse in the seconde place. In the thyrde place thou fyndest 6 and [...] whiche maketh 8, the whiche for bycause it is dygette number, set it vnder that place benethe the lyne. In the .iiii. place is 1 and 9 which maketh 10, and forbycause that this is article nū ­ber, set a cyfer vnder that place benethe the line, and reserue the article to the next place sayenge, 1 that I haue in my mynde and 2 is 3, and 9 is 12, and 9 is 21, and 9 is 30 this is also article number, wherfore set a cyfer vnder that place beneth the lyne, and reserue the article 3 in mynde to the [Page] nexte place. Then come to the .vi. place: sayenge, 3 that I haue in my mynde and 2 is 5, and 6 is 11, this is composte numbre therfore I set the dyget which is 1 ryght vnder that place beneathe the lyne, and reserue the article 1 to the ne [...]te place, sayenge 1 and 1 is 2, and 1 is 3, and 8 is 11, & 4 is 15, this is also composte, therfore sette the dyget 5 vnder the lyne, and adde the article reserued to the figure in the ne [...]te place sayenge, 1 and 3 is 4, and 1 is 5, this is dyget numbre, therfore sette it vnder the lyne, and all is done.

¶Certayne examples to practyse youre selfe in, touchynge the exercyse of Addition.

¶Of the proue of Addition.

¶For the proue of additiō, ye shal make a crosse after the fashyon that foloweth. And then ye shall come fyrst to the addi­ble summes, and plucke out all the 9 that ye fynde there, and the reste what so euer it be, y^t wyll not make 9 set it at the vpper syde of the crosse. Then come to the totall [Page] summe vnder the lyne, and lykewyse deduck all the 9 that ye can fynde there and that that remayneth, not able to make 9 set it [...]t the vndermost parte of the crosse, and yf it be lyke the remenant of the ad­dyble numbres which standith in the vp­per parte of the crosse, your worke was good, yf not it was naught, as by exam­ple ye shall the better perceyue.

¶Of Subtraction, the thyrde parte.

SVbtractiō is a maner of debatyng or subducyng a lesse summe out of a greater: or lyke of lyke shewyng what remayneth.

In subtraction are two numbers, the fyrste is the number abatyd, the seconde, the number abatynge.

¶Then when ye wyll subtrahe any one number out of an other. Fyrste ye shall wryte the number to be abatyd, and vn­der it directely fygure vnder fygure, and place vnder place wrytte the abatoure, and benethe these two summes drawe a lyne, then begyn your subtraction at the fyrste places, and subduce the figure standynge in the fyrst place of the abatour of the fyrste figure standynge in the fyrste place of the nomber to be abatyd: and the [Page] rest that remayneth after the abatement set it ryght vnder the same place benethe the lyne: and so do lyke wyse in the second the thyrd, and al other places. And when ye haue all done, the number that shal re­mayne vnder the lyne, shall be that, that remayneth after the subduction of the a­batour of the number abatyd. As for ex­ample.

¶I lent a man 83456 li. of the which he hath payed me 41111 li. a­gayne now I desyre to know how mych remayneth. Then accordynge to the rule, fyrste I sette the lente money, and ryght vnder that I sette the repayed monie, figure vnder figure, and place vnder place: as ye se by the example Vnder bothe these sommes I must draw a lyne. Begyn to subtrahe the vnder sum out of the vpper, sayenge, 1 out of 6 re­mayneth 5, this 5 that remayneth accor­dynge to the rule set vnder the same place beneth the lyne: then to the seconde place [Page] plucke 3 out of 5 remayneth 2, set that vn­der the lyne: then to y^e thyrd place, plucke 1 out of 4 remayneth 3, set that vnder the lyne: then to the fourth place, take 1 oute of 3 remayneth 2, sette it vnder the lyne: then in the fyfth place, take 4 out of 8 re­mayneth 4, set that also vnder the lyne, & so thou hast fynyshed: Then thou shalte vnderstande y^t it which is vnder the lyne is the remanēt of the monie not yet paid.

¶The proue of Subtraction.

The proue whether ye haue subtrahed well or no, ye muste adde the remayne to the nombre payde, and yf they twayne added together do make the fyrste summe lent completelye then is it well subtra­hyd, yf not, it is not wel subtrahyd, as by the laste example ye maye well perceyue: for by the rule of addition, adde 3 to 7 therof commeth 10, set the sypher vnder the lyne and reserue the article to the next place forth accordynge to the rule of Addition, and thou shalte se this twoo summes ad­ded togyther to come to the fyrste lente summe: and this of Subtraction shalbe sufficient.

¶Of Multiplication.

MVltiplicatiō is a maner of encreasing or augmē ­ting one sum by another In this feat of multipli­catiō are .iii. nūbres to be noted the multiplyed nō ber, the multiplyer, & the numbre that re­downdeth of the multiplicatiō of y^e mul­typlyed [Page] nomber by the multypliar, as in example. Multiply this number 4 by 3 & therof come 12, 4 is the nūber multiplied: 3 is the number multiplyer, 12 the thyrde number that redounded of the multiply­catiō of one of these number by the other then for more experience and redy wor­kynge in this kynde of operation ye shal perfectely knowe by memory the multi­plication of one dyget by an other, the whiche ye shall haue here in this table folowynge, of the whiche one dy­gette ye shall loke for in the hed of the table, and the other in the lefte syde of the table.

¶Here after foloweth the Table.

¶By this table ye shal sufficiently lerne to multyply one dygette by an other. As for example, yf ye wyll multiplye 9 by 5, loke for the 9, at the heed of the table, and for 5 the multiplyer at the lefte syde of the table, then w^t thy fynger descende downe from the place where 9 standeth tyll thou [Page] come before the place where the 5 stādeth and there in the same angle, thou shalte fynge 45, and that cometh of 5 tymes 9, & so do lyke wyse of other.

¶There is also a proper rule for y^e mul­tiplication of one dygette by an other, & it is this, when thou wylte multyply one dyget by an other noote the distaunce of the greater diget from 10 and by the same distaunce multiplye the lesse dyget or e­quall, & that that procedeth of it deducte out of that article whom the lesse number doth denominate, and y^e reste is it that ye seake for, as for example: yf ye wyll mul­typly 7 by 5, fyrste se the distaunce be­twene 7 which is the greater number and 5, and that is 3, by this 3 multiply 5, and that is 15 then subduce this 15 out of y^e ar­tycle that 5 the lesse number doth denomi­nate, whiche is 50, then remayneth 35, that is 5 tymes 7: so lykewyse shall ye doo yf the multyplyar and the multiplyed be lyke. How be it moost ready it is to know without boke very perfytely the multy­plication of euery dyget one in an other.

Now when ye wyll multyply any one number the one by the other. Fyrst wryte fayre your number to be multyplyed, and vnder it the multiplicatour, beneth both these summes, ye shal drawe a lyne. Then shal ye consyder whether your multiplier be a dyget or articly, other elles compost number. I [...] it be dyget number ye shall begynne to multyply by the dyget the fygure or dygette standynge in the fyrs [...]e place of the number to be multiplied, and that that commeth of it, if it be but a dy­gette set it vnde the lyne ryght vnder the same place and thē procede forther to the nexte place and multiply the figure standynge in that place by the same multipli­er, and that that redowndeth of it, yf it be a dygette [...]ette it lykewyse vnder the lyne r [...]ghte vnder the same place, and soo do lykewyse in euery place folowynge, vnto suche tyme as all the figures standynge in euery place, be multyplyed: then that the whiche shalbe founde vnder the lyne is the summe commyng of the multipli­cation of this two nūbers, the one by the [Page] other: as by example ye shall the better perceyue.

¶If ye wyll multyplye this sum 2314 by this 2 [...] ye shall sette your figurs after this sorte, as ye se them. Begyn youre multiplication say­enge 2 tymes 4 is 8, sette that 8 vnder the lyne, then come to the next place and say, 2 tymes 1 is 2, sette it vnder the lyne, then to the thyrde place, 2 tymes 3 is 6, set that vnder the lyne, so to the fourth place, 2 tymes 2 is 4, set that vnder the lyne also, & then thou hast done: so that this number 4 28 vnder the lyne, is it that commeth of the multiplication of this summe 2314 by this number 2. But yf it be so that in the multiplication of any fygure in the number multyplicable, by the multiplyer that it which redoundyth of it be article number, then ye shall set a cypher beneth the lyne ryght vnder y^e same place where the multiplycation is, and reserue the ar­tycle to be addyd to the number that pro­cedeth [Page] of the multiplication of the figure in the nexte place by the aforesayd multi­plyar, the whiche lykewise yf it amounte to an article do likewyse as I byd you to do in the fyrste place: but yf it be number composte, then shall ye set the dyget vn­der the same place beneth the line, and re­serue the article to be addyd lykewyse as is before sayde of article nōber, as in this example.

¶yf ye wyll multy­ply this nōber 8141642 by this fygure 5. Be­gyn at the fyrst place sayenge 5 tymes 2 is 10, nowe forbycause that this number is article, ye shal accor­dynge to the rule before, sette the cypher vnder the lyne, and reserue the article 1 to be added to the nomber that precedeth of the multyplycation of the nexte fygure standynge in the next place of the summe multiplycable, by the multiplyer: so then come to the nexte place sayeng, 5 tymes 4 is 20, to this 20 adde 1 for y^t article that ye reserued, and that maketh 21, therfore by­cause [Page] that this is a composte nūber therfore sette the digette vnder the lyne be­nethe the same place, and reserue the article to the nexte place: then come to the .iii. place sayenge 5 tymes 6 is 30, to this adde the article 2 which ye reserued in the place next goinge before, and then it is 32 sette the dygette vnder the lyne as ye dyd be­fore reseruing the article to the next place then come to the .iiii. place saieng, 5 times 1 is 5 to this adde the article reseruyd, whiche is 3 and that maketh 8, set this digette nōber vnder the line, and then come to the v. place sayenge, 5 tymes 4 is 20, nowe for bycause that this nomber is article set 0 vnder that place beneth the lyue reser­uynge the article 2 to be addyd vnto the next place: then comme to the .vi. place sayenge, 5 tymes 1 is 5, to this adde the ar­ticle 2 reserued and then it is 7, set it vn­der the lyne: thē to the .vii. place, sayenge, 5 tymes 8 is 40, now forbycause it is an ar­ticle nomber ye shall sette a sypher vn­der the lyne, and reserue the article 4 too the nexte place, and for as moche [Page] as there is no mo places, ye shall set this 4 vnder the lyne nexte vnto the 0 that ye sette downe laste, and then ye haue done. ¶When that your multiplyer is com­poste or article, then shall ye take the fyrst fygure of your multyplyer, and by hym shall ye multiplye all the fygures of the multiplycable numbers, settyng alwaye that that amounted of it benethe the lyne as ye dyd before. And when ye haue mul­tiplyed the number multiplycable by the fyrst fygure of the multyplyar: then multiplye it agayne by the seconde figure of the multiplyer, settyng euermore the fyrst fygure of the number multiplycate, dy­rectely vnder the figure multiplycatour, in what place so euer it stande: and the number multiplycable is multiplyed by al the figures of the multiplicatour, then make a stryke vnder them all, addyng al the numbers multyplycate together as they stande, and that which procedeth of that addition is the number multiplyca­ble nowe multyplyed by the hoole nūber multyplycatour, as by this example ye [Page] shall playnly perceyue.

¶If ye wyll multiply this number 2345 by this nūber 1234, set them fyrste as ye se here 2, vnder them drawe a lyne: then begyn with the fyrste figure of the multiplycatour, whiche is 4, and by hym fyrste accordynge to the rule multy­plye all the multiplicable nūber through out, sayenge 4 tymes 5 is 20, sette the cy­pher vnder the lyne reseruynge the arty­cle 2 to the nexte place: then to the seconde place, 4 tymes 4 is 16, to that put youre reserued article 2 and it is 18, set the dy­gette 8 vnder the lyne reseruynge the ar­tycle 1: then to the thyrde place, 4 tymes 3 is 12 and 1 reserued from the place before that is 13, set the dyget 3 vnder the lyne, reseruynge the article 1, then to the .iiii. place, 4 tymes 2 is 8 and 1 reserued is 9, [Page] set that dygette 9 vnder the lyne, and soo haste thou multiplyed this nomber mul­tiplicable by the seconde figure of mul­tiplicatour, Now then accordynge to the rule afore, multiplye the multiplicable nomber by the seconde fygure of the mul­tiplycatour sayenge, 3 tymes 5 is 15 sette the dyget 5 vnder the lyne, accordynge to the rule, which byddeth to sette euermore the fyrst fygure of the number multiply­cate vnder the place where y^e fygure multiplicatour doth stande: as here nowe thou multipliest the multiplicable by the second figure of the multiplicatour, whi­che is 3, thā say 3 times 5 is 15 set this diget 5 vnder the lyne, and beneth the fyrst nomber multiplicate ryght vnder the fygure multiplicatour, as thou seeste in the ex­ample, and reserue the article 1: then to the seconde place of the multiplycable, 3 tymes 4 is 12, and 1 that is reserued is 13 set the dyget 3 vnder the lyne, as ye se in the example, & reserue the article 1, and so to the .iii. place 3 tymes 1 is 9 and 1 reser­ued is 10, set a sypher vnder the lyne & reserue [Page] the article 1: so to the .iiii. place say­enge 2 tymes 3 is 6 and 1 reserued is 7 set it vnder the lyne, thus haue ye done youre multiplication by the seconde fygure of the multiplicatour 3. Then take the .iii. fygure of multiplicatour whiche is 2, and multiply also al the nōbers multiplicable by hym sayenge 2 tymes 5 is 10 set the sy­pher beneth the line right vnder the place where this figure 2 the multiplicator standith, as ye se in the example: and reserue the article 1, then to the seconde place 2 ty­mes 4 is 8, and 1 reserued is 9, set that 9 vnder the lyne: then to the .iii. place, 2 tymes 3 is 6, set that vnder the lyne: so to the .iiii. place sayenge 2 tymes 2 is 4 set that 4 vn­der the lyne. Now begin to multiply with the fourth and last fygure of the multy­plycatour, sayeng 1 tymes 5 is 5 sette the 5 vnder the lyne as I warened ye before, and as ye se in the example, then to the se­conde place 1 tymes 4 is 4, set that 4 vnder the lyne, then 1 tymes 3 is 3, set that 3 vnder the lyne, then 1 tymes 2 is 2 set that 2 vnder the lyne & ye haue done your multiplica­tion: [Page] then muste ye adde accordynge to your rule afore all this syngle multiplied number togyther, and that the whiche cō meth of the addition is the number that cōmeth of the multiplication of this nū ­ber 2345 multiplycable by the number 1234, multiplicatour. Then come to the fyrste place, and se what is there, & there ye shall fynde a 0, set it vnder the lyne, & so to the seconde place: there ye shall fynd 5 and 8 which is 13, set the dygette 3 vnder the lyne reseruynge the article 1 to be ad­dyd to the next place: then come to the .iii. place, there is 0, 3 and 3 whiche is 6 to that adde the reserued 1 and that is 7, set that 7 vnder the lyne, nowe to the .iiii. place, 5, 9, 0, and 9 maketh 23, sette the 3 vnder the lyne, reserue the article 2, so to the .v. place 4, 6, and 7, is 17, to that adde the re­serued 2, whiche maketh 19, sette the 9 vnder the lyne, and kepe y^e article 1 in mynde then to the .vi. place 3 and 4 is 7, and 1 re­serued is 8 set it vnder the lyne, then too the .vii. place, there fynde ye but 2, wher­fore set it vnder y^e lyne, and then haue ye [Page] done: so that this summe vnder the lyne 2893730 is the hole number multiplicate

¶The profe of multyplycacion.

THe profe of multiplicacyon maye be by two meanes. By the subducyng out of all the 9: and the second way is by particion. As concernyng the fyrste waye: ye shall fyrste make a crosse, then beholde the multiplicable nomber, and subdue oute of it all the nynes, and that that remayneth not able to make 9 sette it at the vpper ende of the crosse: then come to the multyplycator, and do lykewise in hym, and that which remayneth all the 9 beynge subducyd, set it at the vnder parte of the crosse: then multiplye the fygure standynge in the vpper part of the crosse by the fygure standyng in the nether part [Page] of the crosse, and out of the same that commeth of it take 9 as ofte as ye can: and that that remayneth not able to make 9, set it at the right syde of the crosse: then come to the totall sum multiplicate, and subduce all the 6 out of hym lykewyse, & that whiche remaineth not able to make 9, set it at the lefte syde of the crosse, and if it be lyke the fygure standinge at the ryghte syde of the crosse, then is it well, o­therwyse it is not well.

¶Of particion the fourth Kynde of Algorisme.

PArticion is a parte of algorisme, by the whiche ye maye easyly diuyde any greater sum by a lesse or equall shewynge howe oftentimes the diuisor is contayned in the nomber diuysyble.

¶In this feate of particion be .iiii. nombers to be noted: the nomber diuysyble, the nomber diuisor, the quotient, and the remayne yf there be any.

¶Before you come to particion it shall be very nedefull and necessarie for you, ryght perfytely to know the table of multiplycacion of dygettes: whiche is set in the chapter of multiplycaciō: For vnlesse that ye know that perfitely ye shal stycke greatly not only in multiplicacion, but also in this feate of particion, and that exactely had in memorye, the rest shall be farre easyar. As for example. yf ye wyll knowe howe often tymes 7 is contayned in 68 ymagyn by & by that this 7 shulde [Page] be contayned 8 tymes: then yf ye knowe without the booke perfytly the foresayde table ye shall se that 8 tymes 7 is but 56 ergo 7 is contayned more then 8 tymes in 68, ymagen then and suppose it to be 9 tymes in 68, then by the table se what 9 tymes 7 is, and thou shalte se that it is 64 wherfore thou mayst conclude that in 68, 7 is conteyned 9 tymes and 4 ouer.

The [...] proue.

¶To diuyde this nomber 45231 by 3 the 3 is diuisor. Fyrste ye shall set downe your nombers to be diuyded, and atthe ende of that nomber on the righte hande ye shall make a streke, wherin ye shall set your quocient, and then set downe youre [Page] diuisor which is 3 vnder the fygure that standeth at the vttermost ende at the lyft hand that is vnder 4, and than say howe many tymes 3 maye I haue in 4, ones 3 and 3 remayneth ouer, set 1 within the stryke and that 1 that remayneth set ouer 4 then stryke the duysor 3 with a dasshe of your pen, and set the dyuisor 3 vnder the fygure 5, then ioyne the artycle 1 to the dygnet 5, and it is 15, then saye howe manye tymes 3 maye I haue in 15, 5 ty­mes 3, set that 5 in the stryke nexte to the fygure 1 and close vp the article 1 and the dyggette 5 with a cyfer 0 ouer eyther of them, and then strike the diuisor 3 with a dasshe of your pen, and set the diuysor 3 vnder the thyrde fygure 2, and se howe many tymes 3 ye may haue out of 2 none therfore sette downe a cyfer 0 within the stryke nexte to the fygure 5 and strike out your diuisor with a dasshe of youre pen, and set the dyuysor 3 vnder the fourth fy­gure 3 then ioyne the article 2 to the dyg­gette 3 and that makethe 23 than se howe many tymes 3 ye maye haue in 23, 7 ty­mes, [Page] and 3 remayneth, sette that 7 within the stryke nexte to the cyfer, and the 2 that remayneth set ouer the fourth figure 3 & close vp the article 2 with a cyfer 0, then stryke out the diuisor and set it vnder the fyrste figure 1 at the ryghte hande, then ioyne the article 2 to the dygget 1 and it makethe 21, than se howe manye tymes 3 ye maye haue in 21, 7 tymes and nothynge remayneth, than set the 7 within y^e stryke and close the article 2 with a cyfer 0 ouer eche of them and strike oute the deuysor with a dashe of your pen, & so the thyrde part of 45231 is 15077.

¶To dyuyde by 2 or 3 fygures, or by as many as pleaseth you.

¶Fyrst set downe your nomber to be di­uyded and your diuisor vnder it, begyn­nynge at the lefte syde at suche a place as ye may take the laste fygure of your de­uysor in the laste ende, and then se howe ofte ye may haue that fygure in y^e figure aboue it, and that set aparte for your quocyent, with the whiche quocient ye shall multiply euery fygure by it selfe of your diuisor and that that cōmeth of the mul­typlycacyon, ye shall abate of the fygure right ouer it, puttynge out that other fy­gure, and set y^e rest aboue it, and so worke with euery fygure by it selfe throughout the diuysor. Then renewe your diuisor 1 fygure forwarde towarde youre ryghte hande, as before is rehersed, and so con­tynue your worde folowynge to the fyrst [Page] fygure of your nomber to be diuided. Then it is to be noted that yf it happe y^t your multiplied number that ye shulde abate be more then the number ouer it, then for a generall rule, ye shall not take your dyuysor out of the fygure aboue it excepte that it maye sufficientlye yelde ynough to all the abatementes of the re­sydue, as more playnly shall apere in the example folowynge.

The [...] proue.

¶Fyrst set downe this nomber 4123, and diuyde it be your dyuysor 12, begyn youre worke at youre lyfte hande, settynge the [Page] article 1 of your deuisor vnder 4, and y^e dygget 2 vnder the thyrde fygure 1, and than se how many tymes the artycle 1 of your diuysor ye many haue in the 4 ouer it, ye wolde saye 4 tymes 1, but that can not be bycause there ye may not haue y^e quocyent 4 multiplyed with the dygget 2 of your diuisor, for therof commeth 8, and then that 8 ye may not take out of 1 ouer the dygget 2 Therefore say againe howe manye tymes 2 maye ye haue in 4, 3 ty­mes and 1 remayneth, set the 3 within the stryke for the quocyēt, & the 1 that remay­neth set ouer 4, and strike out the article 2 of your diuisor with your pen. Then multiplye the quocyent 3 with the dygget 2 of your diuisor, and therof commeth 6. Then ioyne the article 1 that remayneth, & the digget 1, and it is 11 therout take 6 & there remayneth 5, set the 5 ouer y^e thyrde fygure 1, and close vp the article 1 ouer 4 with a figure ouer it, and stryke out y^e dygget 2 of your diuisor againe but one fygure for­warde, as thus: set the article 1 vnder the [Page] thyrde figure 1 is the No. and the diget 2 vnder the second figure 2 & there se howe many tymes 1 I maye haue in 5 that re­mayneth, 4 tymes, & yet there remayneth 1 whiche must be set ouer 5, and stryke out the article 1 with your pen. Then multy­ply the dygget 2 of your deuisor with the quocient 4 and it is 8, then ioyne the arti­cle 1 that remayneth, and the dygget 2 in No. togither & it is 12, then take the 8 out of the 12, and there remayneth 4, set that 4 ouer the seconde fygure 2 in the No. and close vp the article 1 with a cyfer 0 ouer it and stryke out the digget 2 of your di­uysor with your pen. Then renewe youre diuysor againe as before is sayde, & set y^e article 1 vnder the seconde fygure No. and thense how many tymes 1 I haue in 4, that remayneth 3 tymes and 1 remay­neth, set that 3 within the stryke for y^e quocient, & the 1 that remayneth set ouer the 4, and stryke out the article 1 of your dy­uysor with your pen. Then multiply the quocient 3 with the digget 2 of your di­uysor, and it is 6, then ioyne the article 1 [Page] that remayneth, and the dygette 5 in No. and it is 13. Then take 6 oute of 13 and there remayneth 7, set that 7 ouer the dyg­get 3 in the No. and close vp the artycle 1 with a cyfer 0 ouer it, and stryke oute the dygget 2 of your diuysor, and then the 12 parte of 4123 is for the quocyent 343, and the 7 that remayneth shall be sette at the ende of your quocyent, as thus 7/16

[...]

[...]

¶ye shall note that in these two exam­ples the quocient standeth in the myddes betwyxte the two lynes, and the nomber to be deuyded standeth nexte aboue the vppermoste lyne, and the deuysor stan­dethe [Page] nexte vnder that nether lyne. But than ye must marke that there be two dy­uysors, one is called the diuisor currant, bycause it is alwayes remouable toward the right hande in the operation, and al­so it is stryken out, and this dyuysor standethe alwaye vnder the nether lyne of the quocyent. The other diuysor is called the diuysor permanent, for he is not remoued nor blotted as the other is, but standethe alwaye permanent on the lyfte hande dy­rectly againste the nomber that is to be deuided. And iust ouer hym there stādeth the remayne of the whole number whiche remayne can not be deuyded by the deuy­sor, and therfore it is set ouer the deuysor permanent with a strike betwyxte: as ye maye se in the fyrste ensample, where 1 is remaynynge, and 6 is deuysor.

[...]

¶For as muche as in this ensample we [Page] can not take 4 whiche is the diuisor, oute of 3, therfore we shall set 4 vnder 5 & saye howe many tymes 4 haue ye in 35, ye haue 8 tymes 4, and there restethe 3, ye shall set the 8 betwyxte the two lynes, and the 3 aboue 5, then efface the 5 & the 4, then we shall sette, 4 vnder 0 & saye, in 30 how many tymes 4, 7 tymes, set 7 betwene the lynes at the right syde by the 8, and there resteth 2, whiche we shall set aboue 0, and efface 0, than set 4 vnders, and say in 28 howe many tymes 4, 7, & there resteth nothyng, set 7 betwene the lynes by the 7, than set 4 vnder 0 & saye howe many tymes 4 in 0, there is none, there­fore set 0 betwene the lynes, then shall we saye in 9 howe manye tymes 4, 2 tymes, set than 2 betwene the lynes, and restethe 1 which we shall set aboue 9 and efface 9 than 12 howe manye tymes 4, 3 tymes, set than 3 betwene the lynes by 2, & there resteth nothynge. Than in 3 that is the laste fygure howe many tymes 4, no ty­mes therfore at the ende of the figure ye shall set the 3 thus ¾ and it is made.

¶Example whan the diuisor is an arty­cle, it behoueth to do semblably, in say­enge, in 3 howe many tymes 4, no tymes and therfore we shall sette 4 vnder 5, and 0 vnder 0, and saye howe many tymes 4 in 35, 8 tymes, sette 8 betwene the two ly­nes vnder 5, and there resteth 3 whiche we shall sette ouer 5. Then set the 3 that standeth ouer 5 and the 0 together, and that is 30, than saye howe many tymes 4 in 30, 7 and alwayes so to the ende. And than we shall set 4 vnder 2, and 0 vnder 3, and saye in 12 in takynge the 1 that shall rest of the summe afore and shall be aboue 9, and the 2 that is after 9, howe many tymes 4, 3 tymes 4, and thenne set 3 in the number of C, agaynste 2, & than shall we cease, for there remayneth all o­nely 3 to be parted by 40, nowe we shall [Page] not make 0 vnder 3 as is afore, but at the ende we shall set 3 thus 3/40 [...]

¶Ensample whan the diuysor is com­post, as in this fygure afore, we shall say in 35 that ben nere A, howe many tymes 4 that are in the number of B, 8 tyme 4 set that 8 betwene the two lynes in the place of C, and there resteth 3 whiche we shall sett aboue 5 and efface 35 of A, and 4 of B, then shall we say in multiplyenge the 8 of C, by the second fygure of B, that is 2, we shall saye 2 tymes 8 bene 16, than abate 16 of the number of A agaynst the same 42, and there be 3 whiche is ouer 5 and 0 of the number, that be worth 30 and we shall saye, of 30 abate 16 and there re­stethe [Page] 14, of the whyche 14 we shall sette [...] ouer 3, and efface 3, and 4 aboue 0 and efface 0, than shall we set the diuisor somwhat forward, the 4 agaynste 0, that shall be effaced, and 2 agaynste 8, and saye in 14 demonstrynge 1 that shall be aboue 3 and 4 aboue 0 howe many tymes 4, 3 ty­mes, set the 3 benethe the lynes in the nū ­ber of C. and there restethe 2 whiche we shall sette ouer 4, and efface 4, than shall we saye againe in multiplyenge the 3 of C, by the seconde figure of B, that is 2 we shall saye than 2 tymes 3 bene 6 and of 8 that is againste it we shall abate 6 and there shall reste 2, whiche we shall set ouer 8, and efface 8, and alwayes so vn­to the ende, & when we come to the 2 laste fygurs of A, & that we wold deuyde them by 42, we maye not, for the firste that is but 2 shalbe effaced with 1 that standethe aboue 5 and because that we maye take there nothynge we shall set 0 agaynste 2 of A, in the number of C, betwene the ly­nes, and so it is done, & there shall reste 3 to be deuyded by 4 [...], and that 3 shall be [Page] set at the ende of the particion as thus 3/42 and it is finished.

¶And it is to be knowen that as many fygures as foloweth the firste fygure in the number of B, shall be multiplyed by them of the number of C, then the multy­plycacion that therof shall come ye shall abate in the nomber of A, as in this en­sample shall more playnly appere.

[...]

The [...] proue.

¶In this ensample in the number of B, that is the deuysor, be many figures, and therfore we shall saye, in 3 of A, how ma­ny tymes 2 of B, 1 tyme, sette that 1 vpon C and 1 that remaynethe of 3, ouer 3, and than shall we come to the 4 of B and to 1 of C, and multyplye them in sayenge, 1 tymes 4 is 4, whyche 4 we shall abate of the number of A, in takynge 1 aboue 3 and 5 after 3, that shalbe worthe 15 and therof we shall abate 4 and there resteth 11, and for the more shorteste waye of 5 on­lye abate 4, and sette the 1 that remayneth aboue 5, & there restethe alwayes 11, then shall we come to the 3 of B, and to 1 of C, and make all onely the multiplycaciō in saynege 1 tymes 3 bene 3, thanne of 10 abate 3 in demonstrynge 1 ouer 5 and 0 after, and then there resteth 7, whiche we shall sette ouer 0, then bycause of the cy­fer 0 maye nothynge come, we shall leue it, and goo to the nexte fygure and saye 1 tymes 5 that is at the laste ende of B ben 5 but in so muche that we maye nothynge abate of 0 that is againste it in the num­ber [Page] A we shall borowe of the figure afore that is 3 onelye one and efface the 8, and sette the 7 aboue the 8, and the 1 that we shall holde shalbe worth 10 to the regarde of the number that we be in, then we shal saye of 10 abate 5 there restethe 5 whiche we shal set aboue 0, then shall we auaunte our partetour cōsequētly vnder the other fygurs folowynge, that is to saye tyll the laste of B, be set vnder the laste of A, and then ye maye not auaunce them any fur­ther bycause ye be come to the endes of both the numbers.

¶The proue of diuision or particyon is made in this maner: ye shall fyrst make a crosse, as ye dyd before in multiplycacion and abate the 9 of the particyon, and sette the reste at the lyfte ende of the crosse semblably of y^e thyrde nōber that is betwixte the two lynes, and set the rest at the right ende of the same crosse, and yf there be no thynge reste set 0, Then multyply y^e two nōbers of fygure, for they 2 be dyggettes that one by that other, & therof abate all the 9, yf there be nothynge in the fyrste [Page] number, or yf ye may not diuide it ioyne it with the same that shall come thereof. And so the rest that maye not make 9 sette it at the ende vnder the crosse. Then shall we come to the fyrst number & semblably do away the 9 therof, & set the reste aboue the crosse, & if that aboue & that beneth be lyke, the particion is good, or yf not, it is fals. And for to vnderstande it better we make proues by y^e ensamples aforesayde.

proue [...]¶For the fyrst we shall take the parte­tour that is 4 and set it at the lyfte syde of the crosse, than shall we abate the 9 of the thyrde nomber, & there reste 8 whiche 8 we shall sette at the ryght ende of the crosse and multiplye it by 4 and thereof cometh 32 wherof resteth 5 then adiouste them with the 2 farthynges that we myghte not de­uyde, and they shall make 7, the whiche 7 we shall set vnder the crosse, than shal we abate the 9 of the fyrste number that ben the farthynges, and there shall reste 7 [Page] whiche 7 we shall set at the vpper ende of the crosse, and so ben the two endes lyke and it is well made.

¶Reduction.

REduction is a kynde of Algorisme by the which ye be taught to reduce numbers of lesse denominacion or value to numbers of more denominacion or value: or yf the case requyre it, num­bers of great denominacion to the num­bers of lesse value. Example of the fyrst. 20. li. 63. s. 44. d̔. 10. far. Thus reduce the farthinges to pens, & the pens to shelyn­ges, and the shelynges to poundes: and then this summe is 23 li. 6 s. 10 d̔ , and 2 far. so haue you reduced the lesse summe to y^e more. Example to reduce the more to the lesse. Take the same example agayne, and reduce the 20 li. 63 s. 44. d̔. 10. far, all in to farthinges, and it wyll make 22410 farthinges. Fyrst reducynge the pown­des to shelynges, then to pens, & all that pens to farthinges: wherefore it shall be verye necessary for you to knowe what thynge your number dothe signifie, whe­ther [Page] wayght, mony measure, or tyme: and to be expert in all maner of accomptes: it shalbe necessary for you to knowe all maner of wayghtes, coynes, measures, and tyme. Example in englysh money 4 far­thinges make 1 d̔. 12. d̔. maketh a shelynge 20 shelynges maketh a pounde.

¶In weyght, and fyrst of troy weyght, euery pownde hath 12 ounces, and euerye ounce 20 peny weyght, and euerye penye weyght 20 graynes. &c.

¶The haperdepeys pownd hath 16 oun­ces, an ounce 8 drammes, the dramme 3 scruples, the scruple 20 graynes.

¶Of measure, the yarde hathe 3 foot, the foote 12 ynches, the ynche 3 barley cornes of length.

¶Of tyme, the yere hath 365 dayes, the daye 24 houres, the houre hath 60 minu­tes, euery minute 60 secondes, euerye se­conde 60 thyrde, euerye thyrde 60 quar­tes, euery quartes 60 fyftes, euerye fyfte 60, syxtes, and so forth infinitely.

¶To reduce the more summe to the lesse.

¶When thou wylte reduce the more to the lesse, loke howe many tymes the lesse is contayned in the more, and by that nū ­ber multiply the number of the more and that that cōmeth of the multyplycacyon sheweth the more reducyd to the lesse.

Example. I wold reduce 8 d̔. to farthyn­ges loke howe many tymes a farthynge is contayned in a peny, and that is as ye knowe 4 tymes, then multiplye accor­dynge to the rule 8 by 4, and that maketh 32 whiche be 32 farthynges, and so 8 d̔ , ma­keth 32 farthynges.

¶An other example. Here is a summe of 28 li. and 6 s. I wolde haue this pown­des, which is of more denomynacyon re­duced to the shelynges, whiche be of lesse denominacion: then loke fyrste howe ofte a shelynge is contayned in a pound, and that is 20 tymes, for 20 s. makethe a li. multiply then the 28 li. by 20, thereof com­meth 560, whiche be all shelynges: to this putte the other 6 shelynges and so all is 566 shelynges.

¶But ye shall note that where there be any summe of meane denominacyons betwene the more to be reduced and the lesse to whome reduccyon is made: then shall it be easy ar to reduce fyrste the more to the meane, and so by the meane to the lesse.

¶The example. 43 li. 19 s. 20 d̔. 4 far­thynges. yf ye wyll reduce all this sum­mes to the farthinges: then shall it be better for you to reduce y^e powndes fyrste to shelynges and then beynge shelynges to reduce them to pens, and at the laste to farthynges: so by your rule 43 pownde maketh 860 shellynges, to that adde the 19 shellynges, it maketh 879, then reduce this 879 shellynges to pens: loke fyrste howe many pens are contayned in a she­lynge, and that is 12, multyply 879 by 12 therof commeth 10548 whiche be all pens, to this adde your 20 pens, & that maketh 10568 d̔. then reduce this pens to farthynges, se how many farthinges be in a peny that is 4, multyply 10568 by 4 cōmeth 4 [...]272 [Page] to these adde the 4 farthynges and that maketh 42279 farthinges. Thus haue ye reduced 43 li. 19 s. 20. d̔ , 4 farthynges the more by the meane to the lesse.

¶To reduce the lesse to the more.

¶Fyrste marke howe manye tymes the more doth contayne the lesse: and by that number dyuyde the lesse, and the quociēt sheweth the lesse reduced to the more.

Example. I wolde haue this sum 5600 s. reducyd into powndes: for howe manye tymes a pounde doth contayne a shelyng that is 20 tymes, thenne diuyde 5600 by 20 the quocyent shalbe 280, whiche be pown­des: so that 5600 s. reduced to powndes maketh 280 li. & so lykewyse in all other rekeninges.

¶When summes of dyuers denomyna­cyons come in addition to be addyd togi­ther, then begynnynge at the summes of least denominatiō: adde them ouer togy­ther tyll such tyme as they make a num­ber of the nexte denominacion, and that that remayneth not able to make any nū ­ber of greater denominacion, set it vnder [Page] the lyne & procede to the nexte summe of greater denominatiō, to the whiche adde the number of the same denomination reducyd out of the sum before of the lesse denominacion, so procedynge to the ende.

¶Here foloweth of progressyon.

PRogressyon shewethe the nomber whan it begynnethe at 1 or at 2 in mountynge alwayes by one, & one, as dothe this number 123456789.

Now yf ye wyll knowe the valour of the numbers firste ye must regarde two thinges, that is to wyte, yf the nūber procede continually without leuynge any thing betwyxte as here 1234567 or if it leue any thynge betwyxte as here, 13576.

Secondly ye must consider yf the num­ber be euen or odde. And after these two consyderacyous, then by foure rules that here foloweth ye maye knowe the valour of eche whole number.

❧The fyrste rule is whan one number [Page] procedeth in mountinge alwayes conty­nually in the begynnyng, then if it ende in an euyn number, than shal we take the halfe of that euyn number, and by it we shall multiply the odde number that cō ­meth of the euen number, as ye maye se in this ensample folowynge.

¶Here foloweth the rules, and fyrst the rule of thre.

MVltiply by the contrary & diuide by the semblaūt or lyke. This rule maye be vnderstande in two ma­ners. Fyrste multiplye the same that ye wyll bye by his contrary, that is to witte, by the price, and diuide by the semblaunte that is to witte, by as muche as ye haue bought: or thus, multiplye ihe price by his contrary, that is to witte, by the same that thou wylt bye, and diuide it by his semblaunt, that is that same that ye haue bought. And note ye why it is called the rule of thre, for with thre nōbers certayne ye may knowe and fynde the fourth nō ­ber vncertayne. And it is a rule ryght no table and necessarye in the fayct of mar­chaūdyse. For to haue knowlege of this rule, it behoueth to set some rules dyffe­rent in maner of questyons, and fyrke in measures longe.

¶The rule of hole numbers.

¶Yf 9 elles of cloth cost 25 crones, how [Page] moch shall cost 15 by the price. Answer. It behoueth you to set the somme, that is to wyte, 25 crones. And thā ye shal multiply by his contrary, that is to wite, by 15 that ben 375, and than diuide them by that semblaunt, that is to w [...]te, by 9, and therof co­meth 41 crones and an halfe, and there remaineth 1 crone and and halfe, the which ye shall make in sꝪ. and there ben 54 sꝪ. the whiche ye shall deuide by 9 and therof co­meth 6 sꝪ. Therfore ye maye answere that the 15 elles shall cost 41 crones and an half and 6 sꝪ. Nowe yf ye wyll make the proue it behoueth you to forme your question thus yf 15 elles cost. 41 crones and an half and 6 shyllynges, howe moch shal coste 9 elles by the price. Then it behoueth you fyrst to multiplye the 6 sz, by 9 and that bē 54 then it behoueth you to make thereof crones, that is 1 crone an an halfe, & then ye shall multiplye the 41 crones and an halfe by 9, and they ben 373 crones and an halfe, and then set thereto 1 crone and an halfe, and they be 375 crones whiche ye shall deuyde by 15, that ben 25, the which [Page] 25 ben the price of 9 elles, and so the rule is good, and thus ye may do of all other semblable.

☞The secōde rule of hole nombers with numbers broken semblable.

¶yf 10 elles and 2 thyrde partes of cloth cost 35 franc. howe muche shall coste 14 el­les by the pryce. Answer. For to knowe this rule and other semblable: it beho­ueth you to reduce the elles bought, and them that ye wyll bye all into thyrdes bycause of them that be bought, in sayenge thus, 3 tymes 10 ben 30, and set thereto 2 thyrdes, that is than 32 thyrdes. Then it behoueth you to make deuysion by 32, and than ye shall reduce the 14, elles in to 1 thyrde, in sayenge 3 tymes 14 bene 42. Then 42 shall be the multiplicator. Nowe sette the somme, that is to wyte 35 franc. the whiche multiplyed by 42 be 1470 the whiche dyuyded by 32 thereof co­meth 45 franc. and an halfe, and there resteth 14 fran, the which ye shal reduce to [Page] shyllynges, and than deuyde theym by 32 and therof commeth 8 shyllynges, and an halfe, there resteth 8 shyllynges, and than shall ye make them in pens, and dyuyde them by 32, and therof cometh 3 pens, therfore ye maye aunswere that the 14 elles of clothe shall cost 45 francz and an halfe 8 shyllynges and an halfe and 3 pens.

¶For to make the proue it behoueth you to make your worke by the contrarye, for it behoueth you to multyplye the somme that the 14 elles cost by the deuisor, and dyuyde it by the multiplycatour. There­fore sette the somme vppon the lyfte syde, and fyrste multiplye the 3 d̔ , by 32, & whan they be multiplyed ye shal make of them shyllynges, and then ye shall multyplye the 8 sꝪ. and the halfe by 32, and then make therof francz. And then ye shall multyply the 45 francz and the halfe by 32, and de­uyde them by 42, and so ye shall knowe yf the rule be well made.

¶The thyrde rule of hole num­bers with dyuers minutes.

¶yf 4 elles and 2 thyrdes of clothe coste 10 crones, how moch shall coste 6 elles & 2 quarters by the pryce, For to know this rule, it behoueth you fyrst to reduce the 4 elles and 2 thyrdes thus, 3 tymes 4 ben. 12 And than ye shall adioyne the 2 thyrdes, & than it is 14. And than the elles that ye wyll bye, ye shall reduce theym in to one fourthe thus, 4 tymes 6 ben 24. And then set the 2 quarters therto and than there is 26 quarters. And than ye shall multiplye that one by that other, that is to wytte the nombrant of the fyrste by the denomināt of the seconde, in sayenge 4 tymes 14 ben 56. And those 56 shalbe the deuysor. Than multiply the numbrant of the seconde, by the denominant of the fyrste in sayenge 3 tymes 26 ben 7 [...], and those 78 shall be the multiplicator. And therfore set 10 crones and multiplye them by 78, and deuide them by 57. And ye shall fynde that the 6 elles and 2 quarters cost 11 crones and an holfe, 15 shyllynges & 5 pens. And there resteth 8.

¶The fourth rule conteining hole nombers to the marchaundyse that ye haue bought and minutes to the same that ye wyll by.

¶yf 8 elles of cloth cost 15 crownes, how much shal cost two quarters by the price. For to know this rule ye must reduce the 8 elles into quarters, in sayenge 4 tymes [Page] 8 ben 32 then 32 shall be the deuysor, and the 2 quarters shalbe the multiplycator. Now set the 15 crones and multyply them by 2 quarters, and diuide by 32 and ye shall fynde that the 2 quarters coste 0 crones and the halfe 15 shyllynges and an halfe 3 pens. For to make the proue ye must worke the contrary, for ye shall multiply the somme that the 2 quarters coste, that is to wytte, 0 crones, and the halfe 15 sꝪ. and an halfe 3 pens by 32 and dyuyde them by 2.

¶The rule of rounde measures, that is to wyte, measure of corne of wyne and oyle.

¶Fyrste it behoueth you to presuppose and knowe the measures of corne.

¶One muy is worth 12 septiers

¶One septiers is worth 4 mynotes.

¶One mynot is worth 3 busshelles.

¶One busshell is worthe 4 quarters,

¶The measures of wyne.

¶One muy of wyne holdeth 36 septyers

¶The septyer holdeth 4 quarters.

¶The quarte holdeth 2 pyntes.

¶The pynt holdeth 2 choppynes.

¶The choppyne 2 halfe septiers.

¶The halfe septer 2 possions.

¶The fyrst rule.

¶If the muy of corne cost 10 francz, how moche is worth the busshel. Answere. For to knowe this rule ye muste knowe how many busshels ben in 1 muy, Therefore multiply the muy by 12, and than by 4, & than by 3, whiche ben 144 busshelles, the whiche shall be the deuysor of 10 francz therfore deuyde 10 by 144. And therof commeth 1 sꝪ 4, d̔. and an halfe, resteth 24. d̔. Therfore the bushell costeth 1 sꝪ. 4 pens & an halfe, resteth 24 d̔.

¶The seconde rule.

¶To the contrary, yf the busshell cost 1 sꝪ. how muche shall coste a thousand and 4 hondreth Muys by the pryce. Answere, For to knowe this rule, it behoueth you [Page] to make all the Muys in busshels. And there be 201600 bushels, the whiche it behoueth you to multiplpe by 2, and there be 403200, and of them ye shall make crones. Therfore deuyde by 36, and there ben 1 [...]20 [...] crones. Therfore ye maye aunswere that yf the bushell cost 2 sꝪ. a thousaunde and 4 hondreth muys shall cost enleuen thou­sand and 2 hundreth crones, and thus ye may do of all other semblable.

¶The thyrde rule.

¶yf the septier of corne be worth 1 francz & the lote of peny torneys weyght 12 oun­ces, how mych ought it to weygh whan y^e septer is worth 15 tornoys. Answere. Multiply the fyrst nōber by the seconde, that is to wytte, 20 by 12, and deuyde it by 15, & ye shall fynd that it ought to wegh 16 ounces. And thus ye may do of al other lyke.

¶If the muy of wyne be worth 12 francz howe muche ought the pynte to be worth Answere. For to knowe this questyon, it behoueth you to reduce the 12 muys into septyers, from septiers into quartes, and [Page] from quarters into pyntes, and that ben 188 pyntes. And than ye shall reduce the 12 francz in to sꝪ. that ben 240, and that into pens, that bene 2880 pens, the whiche behonethe you to deuyde by 288 & it cometh to 10 d̔. Therfore yf the muye of wyne cost 12 francz, the pynte is worthe 10 d̔. But it is requisite that the Tauer­ner haue some gaynes yf ye sell 12 d̔. the pynt. I demaunde howe moche shall he wynne vpō the muy: Answer. He selleth it 2 d̔. more than it is worth therefore mul­typly 288 pyntes by 2 and they be 576, the whiche ye may deuyde by 12 and ther shal be 48 sꝪ. Therfore may ye answere that he getteth 48 sꝪ. vpon the muy.

❧yf the muy cost 10 francz how mych is worth the pynt. Answere. It behouethe you to do as is aboue sayde: and ye shal fynd that it is worth 8 pens and 1 thyrde. ¶yf the pynt cost 6 pens, how much shal cost 12 muys by the pryce. Aunswere. It behoueth you to know how many pyntes ben in a muy, that is 288, multyplye 12 muys by 288 that is 3456, pyntes. And [Page] than multiply the pyntes by 6 that bene 207, 6, of whome ye shall make sꝪ by diuy­sion. and there ben 10728 sz, and of shyl­lynges ye shall make francz. Therefore ye shall make diuysyon by 26, ye shall fynde 86 francz 8 sꝪ. Therefore ye maye aunswere that the 12 muys shall coste 86 francz 8 sꝪ.

¶In so myche as competently we haue tracted of the rule of thre in the fayct of measures, it is expediente that we tracte therof in the faycte of weyghte.

¶yf an hondreth poundes of peper cost 20 sꝪ. how muche shall cost 6 pound by the price: Answere. For to knowe this que­stion, ye must multyply by the contrarye and deuyde by the semblaunt, that is to wyte, multiplye by 6 and dyuyde by 10, and ye shal fynde that the 6 poundes shal cost 1 francz, and 4 sꝪ. To make the proue ye must multiplye by 100 and dyuyde by 6. Now I demaunde of the 6 poundes cost 1 franc 4 sz, how moche is worth the ounce. [Page] For to knowe this ye shal make the poundes in ounces, the whiche ben 96 ounces, & then make the money in pens, the which ben 288 d̔. the whiche ye shall deuyde by 96 and therof commeth 3 pens, therefore the ounce shall coste 3 pens.

¶If one li of saffron cost 3 francz and an halfe, how muche is worth the ounce. Answere. It behoueth you to knowe that in a pounde ben 16 ounces, therfore deuyde the 3 francz, and the half by 16 and ye shal fynde that the ounce is worth 4 sꝪ. 4 d̔. and an halfe, & thus ye maye do of other lyke. ¶If 4 pounde of saffron cost 16 fran. 6 sꝪ 8 d̔. howe much shall cost 3 quartrones by the pryce. For to knowe this rule, ye shall reduce the 4 li. in thyrdes and shall saye [...] tymes 4 ben 12, and 1 thyrde ben 14 then ye shall multiply by 4, and shall say 4 tymes 14 ben 56 the deuysor, than for the seconde number we shall say, 3 tymes 3 ben 9 four­thes or quarters, the whiche 9 shall be the multiplyer. Now set the 16 francz 6 sꝪ 8 d̔. tournoys, & multiplye them by 6 and de­uyde them by 16, and therof cometh 2 fran. [Page] and an halfe, 2 sꝪ. 6 d̔. therfore ye may an­swere that the 3 quarters shal cost 2 fran. & an halfe 2 sꝪ. 6 d̔. For to make the proue ye must worke by the contrary in multy­plyenge by the diuysor, that is to wyt by 56 and make diuysyon by 9, and so maye ye do of other semblable.

¶If one poūd of tyn cost 9 blances, how many hundreth shall I haue for a thou­sand and 4 hundreth francz. It behoueth you to knowe how moch is worth y^e hun­derth by 9 blances the pound. And ye shal fynde that there is 12 francz and an halfe. Now make diuysion of 1400 frances by 12 frances and an halfe, ye shall fynde 112. Therfore ye maye saye that I shall haue 112 pound of tynne for 1400 frances, And also as we haue made this rule, ye maye do in all other marchandyses, as in lead, yron, spyces peper, suger. And as we haue done of poundes ye maye do of quartrons, ounces, & al other weyghtes,

¶A rule whiche is without tyme

THre marchauntes put theyr monye togyther for to haue gaynes, y^e whi­che [Page] haue boughte suche marchaundyse as hath cost 125 francz, whereof the fyrste hath laide 15 francz. The seconde 64 sꝪ. and the thyrde 36, fꝪ. And they haue go­ten 54 franc. of clere gaynes. I demaund how shall they deuide it, so that eche man haue gaines accordynge to the moneye that he hath layd downe. Answer. In all suche rules and questions ye shall multiplye eche one after the money that he had layde, therfore multiplye the gaines for the fyrst by 25 and deuide by 125, that is the diuisor commune. For the seconde mul­tiplye the gaynes by 64, and dyuyde by 125 the diuisor commune. And for the thirde multiply the gaines by 36, and dyuyde 125 the deuysor commune. And for to fynd the deuysor commune, ye shal set togither the multiplycatours, that is to wyte 25, 64 and 36 which is 125 the deuysor commune. And so shal ye do in al rules of company Nowe ye maye fynde & knowe how moch eche one hath of gaines, and ye may se it by the ensaumple here present.

The fyrste hathe 10 fꝪ and halfe 2 sꝪ. The [Page] seconde hathe 27 fꝪ & halfe 2 sꝪ. and halfe 5 d̔. & halfe, resteth 2 d̔. and halfe, The third hath 15 fꝪ and halfe 5 sꝪ. resteth 60 pens.

And they haue yet to be deuyded among them of restes 62 d̔. and and halfe.

¶For to make y^e proue it behoueth you to diuyde the restes, and than reduce all togithers, and ye shall fynde the somme diuided, for all the rules of company ben proued by addicion of sommes.

¶The seconde rule of hole tyme.

¶Foure marchauntes laye money together for wynninge for a certaine tyme, of whome the fyrste hath laide 10 fꝪ. for two yere. The second 20 fran. for 3 yere. The thyrde 100 francz for one yere. And the fourth hath laide 40 franc. for 4 yere: and they haue gayned 454 fran. I demaunde how moch eche one ought to haue of winnynge after the money that he hath layd, & after the tyme that he hath holden his money in gayne for company. Answere. [Page] For to knowe this rule and other sem­blable, ye shall multiplye the monye that eche one hathe layde by the tyme that he hath holden it in companye. Example. The fyrste hath layde 10 fran. for 2 yere, therfore it behoueth you to multyplye 10 by 2 in sayenge 2 tymes 10 ben 20. For the seconde 3 tymes 20 ben 60. For the thyrde 1 tyme 100 is an 100. For y^e fourth 4 tymes 40 ben 160. & then it behoueth you to fynd a dyuysor cōmune, for eche hath his mul­tiplycator, that is to wytte, the same that he hath layde, and for to fynde it ye shall sette together all the multiplycatours, that is to wytte the 20, 60, 100, 160 the whi­che maketh 340, therfore these 340 shall be the dyuysor commune to all, thenne howe muche eche one oughte to haue ye maye se by the ensample here folowynge 454 sꝪ. The fyrste hath 26 francz and halfe 4 sꝪ. one peny, resteth 140. d̔ ,

The seconde hath 80 frances 2 sꝪ 4 d̔. reste 80 pens

The thyrde hath 134 frances 1 sꝪ. 5 pens, reste 20 pens.

The fourth hath 213 franc. and an halfe 2 sꝪ and a halfe 5 d̔. rest. 100 pens.

Of rest they haue to deuide one peny.

¶The rule of company where as is hole tyme and partes of tyme.

THre marchauntes lay money in cō ­pany for to haue gaynes thereby, of whome the fyrste hathe layde 30 frances for two yeres. The seconde hath layde 400 fran. for one yere & thre monethes. And the thyrde hath layde 60 fran. for thre ye­res & two monethes. And they haue gay­ned with this money 44 franc. I demaūd howe they shal deuyde it to the ende, that eche one haue his right after the money and the tyme that they haue sette and holden for to gayne. Answer. For this rule & all other semblable, ye shall multiply the tyme by y^e money, as we haue saide aboue but for as much as there be monethes ye must set & reduce all the tyme of eche one in monethes, and also yf there were anye [Page] dayes ye shulde set all the tyme in dayes. The fyrste hathe layde 30 frances for 2 yeres, in 2 yeres ben 24 monethes, there­fore multyplye 30 by 24 there bene 720, and these 720 shalbe the multyplycatoure of the fyrste. The seconde hathe layde 40 fran. for 1 yere and 3 monethes, in one yere ben 12 monethes, and 3 dothe make 15 monethes, multyplye 40 by 15, they make 600 whiche is the multiplycatoure of the seconde. The thyrde hathe layde 60 francz for 3 yeres and 2 monethes, 3 yeres ben worth 36 monethes and 2 bene 38 monethes. Nowe multyplye 60 by 38, and there ben 2280, which shall be the multiplicatour of the thyrd. Now for to haue a dyuysor comune, ye shall set togyder all the multiplycatours that is 3600 the dyuysour comune. They haue to deuyde 44 francz. The fyrste hath 8 francz, and halfe 6 sꝪ rest [...]. The seconde hath 7 frances. 6 sꝪ. and half. rest 0. The thyrde hath 27 francz and halfe 7 sꝪ. 4 pens, rest 0.

¶A rule of dyuers syluer and dy­uers tyme.

THre marcaūhtes haue made companye togyder of whom the fyrst hath layd 10 francz. 4 shyllynges for 2 monethes. The seconde hath layd 15 frā. for one yere. And the thyrde hath layde 6 francz 7 sꝪ for 8 monethes, and they haue goten of this money 24 francz. How they shall dyuyde it after the money and after this tyme I demaunde. Answere. For to knowe this rule and all other semblable it behoueth you to reduce the money of euery man in shyllynges. And all the tyme in monethes. And then multiplye the money by the tyme. Ensample. The fyrste hath layde 10 francz that ben 200 sꝪ. and 4 ben 204 the whyche ye shall multy­plye by 2 monethes, and they shall be 408 the multiplicatour of the fyrste. The se­conde hath layd 15 francz for one yere, and in 15 francz ben 300 sꝪ. and in one yere ben 12 monethes, therefore multiplye 300 by 12, and there shall be 3600 the multyplye­catour of the seconde. The thyrde hathe [Page] layde 6 francz 7 shyllynges, and in 6 frā. ben 120 sꝪ. and 7 ben 127 sꝪ. for 8 monethes, therfore multiply 127 by 8, and they shall be 1016 the multiplicator of the thyrd. And for to haue the deuysor comune, ye muste reduce togyther all the multiplicators, & that shall be the deuysor commune, as ye may se by the example folowynge. They haue 24 frances of wynnynge.

The fyrst hath 2 francz and halfe 8 sꝪ. and halfe 5 pens and halfe restethe 1360 pens. The seconde shall haue 17 francz, 3 sꝪ. and halfe d̔. resteth 1952 pens.

The thyrde shall haue 4 sꝪ. & halfe, 7 sz, 0 pens, & halfe resteth 17112 pens.

And they haue to deuyde 1 d̔. of the restes For to make the proue ye shall reduce togyther the thre sommes that they haue had. And yf there be more or lesse the rule is euyll made.

¶Here foloweth the rule of com­pany of factours with mar­chauntes seruaūtes.

OF this rule of factours ye may ma­ke 3 rules in maner of questyōs that fall amonge marchauntes. Example, 8 marchauntes 5 factours, and 3 seruaun­tes or varlettes haue made companye togyther, and haue clerely gotten 150 fran. wherof the factours oughte to haue the halfe of the marchauntes, and the seruaū tes the thyrde parte of the factours, how shall they dyuyde these 150 franc. Answer For all suche rules and questions it behoueth you to fynde a nomber wherin is an halfe and a thyrde, and that shal be 6, and these 6 shalbe for the marchaunt. And the halfe of 6 ben 3, that shall be for the fac­tours, and the thyrd part of the factours is 1 whiche shall be for the seruauntes. And than ye shall multyplye the one by y^e other, that is to wytte, the personages by theyr nomber, 6 tymes 8 ben 48, and these 48 shall be the multiplicator of the mar­chauntes. And than there ben 5 factours, that haue 3 and 3 tymes 5 ben 15, and than there ben 3 seruauntes that haue 1, and 1 tymes 3 is 3, & therfore the factours shall [Page] ye multiplye by 15 and the seruauntes by 3. Nowe for to fynd the dyuysor comune ye shall set together all the multyplyca­tours, that is to wyte, 48, 15, 3, whiche bene 66 these 66 shall be the dyuysor commune. Example they haue to dyuyde 150 francz. The marchauntes haue 109 franc. 1 sꝪ. and halfe 3 d̔ , and halfe, resteth 21 d̔.

The factours haue 34 fran. 1 sꝪ. and halfe 3 d̔. and halfe, resteth 21 d̔.

The seruauntes haue 6 fran. and halfe 6 sꝪ. 4 d̔. rest 24 pens.

¶They haue to deuyde 1 peny of restes For to make the proue ye shall dyuyde al the restes by the dyuysor comune. And than ye shall reduce all togither, for to haue 150 frances.

¶ The rule of factours the which gate the halfe of the gayne and of the princypall.

¶And other rule in maner of a question a marchaunt hath gyuen 50 franc. to his factor by suche couenant that he gouerne [Page] them for 10 yeres. And at the ende of the tyme, that is to wyte, at the ende of 10 yeres. And at the ende of 10 yeres, they shall dyuyde the gaine and the principall It hapneth that the factour wyll go his waye at the ende of 6 yeres, and he fyn­deth that he hathe gayned a thousande francz. I demaunde how ought the sayde factour to be paied, and how moch ought the said marchaunte to haue: Answere. ye ought to regarde howe moche he shold haue gained in those 10 yeres that he sholde haue holden them in gayne as he had promysed. Therfore ye maye forme the questyon, yf 6 haue gotten a thou­sande: howe muche shall be the gaynes of 10. Multyplye 1000 by 10 and dyuyde by 6 and ye shall fynde that he shoulde haue gutten 1666 fran, and an halfe 3 sꝪ. 2 pens. Of the whiche gaynes the mar­chaunt ought to haue the halfe that ben 833 francz. 6 shyllynges and halfe and 1 peny. And than take vp those 833 fran. 6 shylynges and halfe 1 penye of 1000 francz that he hath gayned, and there re­mayneth [Page] 166 frances, 13 shyllynges 5 pens for the factour. Nowe ye maye aunswere that the marchaūt shall haue of the gay­nes 833 francz 6 shyllynges, and halfe 18. And the halfe of the princypall, that is to wytte of 50, that is 25 and there ben 852 francz. 6 sꝪ. and halfe 1 d̔. And the factoure shall haue of gaine 166 francz. 13 sꝪ. 5 pens. And of the princypall 25 that ben 191 fran. 13 sꝪ 5 d̔. And thus maye ye do of all other semblable. And it is proued by the reductyon of the two sommes gayned.

☞ The thyrd rule of factours with couenauntes, y^t the factour shall gayne the halfe of the princypall.

¶An other rule of company of factours & marchauntes with conuenaunt that the factours shall gaine the halfe of the princypall and not of the gayne. Example. A marchaunt gyueth vnto his factoure 400 fran. that he shall gouerne them for 6 yeres, & at the ende of the tyme the halfe of the princypal shal be to the factour. It happeneth the factour wyll go his waye [Page] at the ende of 2 yeres, & hath gayned 200 frā. I demaunde how ought the factour to be payed. Answere. ye ought to regard howe moche he sholde haue gayned yf he had serued all his tyme, and for to fynd it ye maye worke by the rule of thre, for ye must multiplye by his contrarye, that is to wytte by 6, and diuyde by his semblaūt that is to knowe, by 2, in saienge yf 2 haue gayned 200 frances: howe muche shall 9 gayne, and ye shall fynde that he should haue gotten 900 franc, and he gayned but 200 franc. wherefore he oughte to make a gaine 400 fran. to the marchaunt: and he ought to haue the half of y^e principal, that ben 200 frances, therfore he oweth 200 vn­to the marchaunte, and so he hath loste all his tyme, and 200 fran. of aduauntage for the marchaunt ought nothyng to lose lyke as he had accomplyshed all his time

¶The thyrde rule of chaunges for to vse deceyte or fraude.

¶Two marchaūtes wyl chaunge theyr marchaūdyse, & the one begyled the other [Page] the one hath peper, and that other cloth. He that hath peper wyll sell for 25 franc. the hondereth by chaunge, whyche is no more worthe than 20 fran. in syluer con­tented. I demaunde for how moch ought the other to sell vnto him the elle of hys cloth, that is worth but 15 sꝪ. to kepe hym selfe from losse. Answere. For the rule of thre ye maye saye thus, yf 20 frances of content gyue me 25 fran. at the chaunge howe muche shall gyue me 15 of contente. It behoueth you to multiplye the 25 by 15. whiche ben 373, the whiche ye shall dy­uyde by 20 and therof commeth 18 sꝪ. 9 d̔. therfore ye maye say that he shall sell the elle of clothe for 18 shyllynges 9 d̔. And thus may ye do of all other.

¶ Two marchauntes wyll chaūge theyr marchaundyse, of whom that one hath 100 pounde of wolle, that is no more worthe but 15 crones. And he wyll chaunge wyth an other in a pyce of cloth that is worthe 21 crones, and he wyl gyue hym the wolle for 17 crones. I demaunde for how moche ought the other to sell the peyce of clothe [Page] to the ende that he be not betromped [...] Answer. By the rule of thre whan 15 are worth 17 demaunde howe moche shall be worth 21 Dyuyde by 15 and ye shall fynde the same that ye requyre.

¶ Two marchantes wyl chaunge theyr marchaundyse, and the one defraude that other that hath peper, and wyll sell it 24 fran. the hondreth by chaunge. whiche is no more worth but 20 frances in moneye content, and he wyll haue the halfe in money contente. I demaunde for howe mych ought the other to sel the elle of his cloth that is no more worth but 15 sꝪ. An­swere, ye must take awaye the money content that the other demaundeth, that ben 12 franc. for the iust price, & of the whiche he wyll sell ouer. Therefore take awaye, and withdrawe 12 of 20 franc. which is the iust yrice, and there rest 8 fran. for 8 and 4 ben 12, And ye maye saye by the rule of thre, yf 8 gyue me 12, what shall gyue me 15 sꝪ. whiche is the iuste pryce of the clothe, multyply 12 by 15 and dyuyde by 8, and therof cometh 22 sꝪ. 6 d̔. And therefore [Page] the marchant ought to sell the elle of his cloth after 22 sz, 6 d̔. els he shold haue losse And thus ye ought to do of al maners of chaunges and barathes, for yf he y^t hath the peper, demaūded but the thyrde or the fourth or 2 or 3, abate all onely the same y^t he shal demaunde, and then by the rule, as is said. And note ye well that yf he wil multiplye shyllynges, ye shall haue shyllynges. And of crones ye shall haue cro­nes, and of frances ye shall haue frances And in lyke maner of all other.

¶Here foloweth many rules & questiōs to haue the more knowledge of y^e science of arismetryke, and the fyrste is of col­lectes and tallyages.

TEnne men owe vnto the kynge of collecte and tallyage 244 fran. I de­maūde howe shall they diuyde them to the ende that eche one paye after the valour of his goodes, for it is reason that more be payed by the riche then by the poure. For he that is more endowed with goodes is more holden vnto god and to [Page] the prince. Answer. It behoueth to know how muche eche one is worth in his goo­des, and in his possessions.

• The fyrste is wourth 100 francz
• The seconde is wourth 400 franc.
• The thyrde is wourth 154 franc.
• The fourth is wourth 1000 franc.
• The fyfte is wourth 1150 franc.
• The syxte is wourth 40 franc.
• The seuenth is wourth 440 franc.
• The eyght is wourth 80 franc.
• The nynth is wourth 600 franc.
• The tenth is wourth 360 francz.

Nowe it behoueth you to fynde the mul­typlycator and the deuisor. The multy­plycator shalbe eche one by hym self, and so for the fyrste it behoueth you to multy­plye by 100, for the seconde by 400. for the thyrde by 154, and so must ye do of the o­ther: And for to fynde the diuisor, ye shall sette togyther all the multiplicatours, as 100, 400, 154 &c. and all that togither shall be the diuisor commune, whiche is 4464 Therfore multyply the collecte, that is to wytte, 244 for eche one his valour, and dyuyde [Page] by 4464, or by the halfe that is 2232, and then ye shall wryte howe moche eche one ought to paye. Example.

The fyrste sholde paye 5 franc. 9 shyllyn­ges 3 d̔. and halfe reste 1464.

The seconde sholde pay 12 franc. 17 shyllynges 3 pens, re [...]eth 1 [...]92.

The thyrde sholde paye 8 frances 8 sꝪ. 4 pens resteth 660.

The fourth sholde paye 54 frances 15 shyllynges 2 pens, resteth 1248.

The fyfte sholde paye 62 frances 17 sꝪ. 2 pens resteth 96

The syxte sholde paye 2 frances 3 sꝪ. and halfe 2 pens and halfe, resteth 1032.

The seuenth shulde paye 24 frances 1 shylling 0 pens, resteth 192

The eyght shoulde paye 4 frances 7 sꝪ. 5 pens resteth 2064.

The nynth sholde paye 23 frances 15 sꝪ. 10 pens and halfe, resteth 2088

The tenth shoulde paye 27 frances 6 sꝪ. 7 pens resteth 624

¶And they haue to dyuyde 2 pens and halfe of restes. Then whan ye haue all [Page] dyuyded and wryte the somme and the restes, ye shall set togyder all the restes, and diuide them by the deuysor comune, or by the halfe. And yf there be more or lesse. the rule is not well made, for the re­maynant of all ought to be dyuyded by the diuisor comune. And the proue of this rule is reduccyon. And marke well this rule for it is right good vnto the coun­tre where all the goodes be praysed by al the townes and castels, as it is in manye places of Daulphyne, and of Prouence

¶The rule of the mylnes.

¶One man hath thre mylnes of whom one gryndeth eche day 5 septyers of corne and the other gryndeth 7 and the thyrde 8. There commeth a marchaunt that wyll haue gronden one hundreth septyers of corne, I demaund how ought the mylner to dyuyde the corne to the mylnes to the ende that eche one haue assone done as an other. Answere. For to know this questyon and rule. ye must fynde the dyuysor and the multiplicator, the multyp. shall be eche one by him selfe, and the deuysor [Page] shalbe the thre multiplycatours set togy­ther y^t ben 20. Therfore yf ye wyll knowe howe muche corne ought to be layde vp­pon the fyrste mylne, ye muste multyplye the 100 septiers of corne by 5 & deuyde by 20, whiche shall be 25 septiers, that shalbe layde vppon the fyrst mylne. And for the seconde ye shall multiply 100 by 7 and dy­uyde by 20, and there shall be 35 septiers, y^e whiche ye shall putte vppon the seconde mylne, and for the thyrde ye shall multy­plye 100 by 8 and dyuyde by 20 & there shal be 40 septiers, whiche ye shall put vppon the thyrde mylne. And thus may ye do of all other semblable. It maye be made o­therwyse, set togyther the sommes that y^e thre mylnes grynd that is 20, and by the rule of thre ye shall say, yf 20 gyue me an 100, howe moche shall gyue me 5 or 7 or 8. And it is proued by addition. Example. 100

The fyrst shall haue 25 septiers. The se­conde 35 septiers. The thyrde 40 septiers,

The rule and questyon of a shepeherde or pastour.

FOure men haue 300 shepe or mou­tons, of whome the fyrst hath an 100 shepe / the seconde 40, the thyrde 150 and the fourthe 10 And they gyue vnto a shepherde for to kepe these shepe 25 fꝪ. for a yere. I demaūde how ought the one to paye of the 25 fran. after the shepe that he hath. And how long time ought eche one to haue hym at comense or meat. Answer. For to knowe this rule and al other sem­blable, it behoueth you to fynde y^e multyplycator and the deuysor, the multyplyca­tor of the fyrste shall be 100, of the seconde 40, of the thyrde 150, and of the fourth 10, & than set togyther all these som̄es the whi­che ben 300 the dyuysour comune. Or ye may make it by the rule of thre in sayeng, yf 300 gyue me 25, howe moche shall gyue me 100 or 40 or 150 or 10, & alwayes dyuyde by 300 and thus of all other rules.

¶The rule and question of zarasins for to cast them within the see.

THere is a galle vpon y^e see wherein be thyrty marchaūtes, that is to wit 15 crysten men, and 15 sarazyns, ther falleth great tempest where vpon it behoueth them to cast all the marchaundyse in to the see, and yet for all that they be not in surete from perysshynge, for the galle is feble and weke, so that by ordynaunce made by the patrone, it is necessary that there be caste into the see the halfe of the thyrty machaūtes, but the sarazyns wyll not be cast in, nor also the christiens: then by an apoyntment made, they shall sette them downe vpon a rowe. & then counte them vnto 9 and he that sholde fall vpon the 9 to be caste into the see, how wolde ye [Page] set them that none of the chrystyens shold be caste into the see. Answer. ye shall or­deyne them after these meters folowyng. Post .iiii. qui (que) da post duos vnū colloca Tres numerabis, postea vnū collocabis Vnū dic panther, & duo consequenter, Duos post ponas &, iii. siml hic apponas Semel dic an̄ bis. post .ii. vnū terminabꝭ Primi christiani, sunt saraceni (que) secundi. That is to wytte, 4 christiens 5 sarazins 2 christyens 1 sarazyn, 3 christyens 1 sarazin [...] christien 2 sarasyns, 2 christiens 3 sara­zyns, 1 chrystyen 2 sarazyns, 2 christiens 1 sarazyn. Or for to know it more shortely ye may worke by this verse folowynge, by the nomber of the vouels.

Populeam virgā matrē regina tenebat

¶The rule and question of a testament.

A Man hath made his testament, the which hath lefte his wyfe great, and hath ordeyned in his testame [...]t that yf she brought forth a sone, he shold haue two partes of his goodes, that is to wyt, of 1200 crones, and his wyfe y^e other part, [Page] and yf she brought forth a doughter, then the moder shold haue two partes, and the doughter the other parte. It happeneth whan the man is dede, the wyfe bryngeth forthe a sone and a doughter. I demande how shall they deuyde the 1200 crones. Answere. ye shall set 1 for the doughter, and 3 for the mother, for y^e mother ought to ha­ue two partes agaynst the doughter, and set 4 for the sone for he ought to haue two partes agaynste the mother. Therfore ye shall multiply the 1200 crones by 4 for the sone, by 2 for the mother, and by 1 for the doughter. And for to fynde the dyuisor ye shall set togyther 1, 2, and 4, whyche ben 7, therfore dyuyde by 7. Example.

• 4 The sone shall haue 685 crones & an halfe, 7 sꝪ. 8 d̔. & halfe, resteth a halfe d̔
• 2 The mother shall haue 342 crones, & and halfe 12 shyllynges and halfe, 4 d̔ Resteth 2 pens.
• 2 The doughter shall haue 171 crones. 15 sꝪ. 5 pens Resteth 1 d̔. 7

Multiplicators Deuysor.

They haue to dyuyde an halfe peny.

¶The rule and question for to buylde. And fyrste for the place.

A Man hath a groūd that is in length 100 yardes, and in brethe 70 yardes, where as he wyll edyfye and buylde [...] houses, of lengthe 5 yardes, and 4 brethe I demaunde how many houses shall he haue vpon that ground. Answer. ye shall multyply the lenghte by the breth in say­enge 70 tymes 100 ben 7000, an be [...]he house must haue 5 yardes of length, and 4 of brede / multyplye that one by y^e other, and they make 20, whiche 20 shall be the dyuy­sor cōmune, therfore deuyde 7000 by 20, & ye shall fynde that there shall be 350 hou­ses Note well this rule.

¶The rule & questyon of the walles.

A Manne wyll make a wall 32 fote in lenghte, and 2 of thycknes, and the heyght 25 fote, and eche fote shal cost the makynge 2 sꝪ. I demaunde how moch shall cost the makynge of all the wall. Answere. For to know this rule, ye shall multyply the lengthe by the thyckenes in [Page] sayenge 2 tymes 32 ben 64 / & then ye shal multyplye it by heyghte in sayenge 25 ty­mes 64 ben 1600, and than multyply by the pryce, that is to wytte by 2 shyllynges the which ben 3200 shyllynges, wherof ye shal make francz, therfore dyuyde them by 20 and they ben 160 francz. And so moch shal coste the makyng of the wall.

¶The rule and questyon of the couerynge.

¶If ye wyll haue a house couered with tyelles, ye must knowe how many tyelles behoueth you to haue vnto the length of a lygne, and how many to the bredde. Example. If the house hadde nede of 54 for the length, and 34 for the brethe, I demaunde howe many shold be requysy [...]e vnto all the house. Answere. Multyplye the length by the brethe in sayenge 34 ty­mes 54 ben 1836 tyelles, and so many must ye haue to couer the house.

The rule & questyon of a graden.

¶A louer dyd entre into a garden for to [Page] gather apples for his lady, and vnto the sayd gardyn ben thre gates, and in eche gate is a porter, and whan he shall ysue after that he hath gathered the apples, he must gyue the halfe of his apples & one, to the fyrste porter, and whan he is at the second porter, he must gyue vnto hym the halfe and one / and to the thyrde porter y^e halfe and one, & whan he is forth he hath no more but one apple to gyue vnto his lady paramour. I demaunde how many apples had he gathered. Answere. He had one, aple whan he was forthe, set to it one, and than it is 2, and then double the 2 and it is 4, therfore he hadde 4 at the thyrde porter. Then to this 4 set 1 & that is 5, and then double them and that is 10 therfore he hath 10 apples at the seconde porter, to this 10 sette 1 and it is 11, double them, and that be 22 apples. Therfore ye maye say that he had gathered 22 apples.

¶The rule and questyon of a ladder or stayre.

¶I haue sene a stayre that had 100 step, [Page] pes, in the fyrst steppe was 1 douffe in the seconde step 2, in the thyrde 3, in the fourth 4, and so vnto 100, I was demaūded how many douffes were in al the stayre. I an­swered 5050. Probacyon I wyll gyue you certayne of all nomber that do procede naturally, that is to wyt, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, And infinitly as ye wyll, for all nō ­ber naturall is ended in nomber euen or in nomber not euen, yf it be ended in nomber euen, than by the halfe therof multy­ply the nomber not ouen, that encloseth it Example. 1, 2, 3, 4, wyll ye know what all amounteth vnto in sayenge 2 tymes 5 ben 10, for 2 is the halfe of 4, and 5 is the nom­ber not euen that encloseth 4, And yf the nomber ende in nomber not euen. As by ensample, 1, 2, 3, 4, 5, wyll ye knowe what all amounteth vnto. Multyply 5 by his greater halfe, that is 3, sayenge 3 tyme 5 ben 15, And thus shall ye alwayes doo in what nomber so euer it be euen or not euen &c.

¶The rule and question of two men.

IF two menne go by one waye, and that they go in to any farre place, and procede in such wyfe, that the one procede eche daye certayne nomber of my les, that is to saye 4 and 6 more or lesse. And that other man goeth encreasynge the fyrste daye one myle, the seconde daye two, the thyrde thre, and so encreasynge after progressyon. Be ye all certayne that in some day the one ouertaketh the other. It is demaunded in what daye, and how many myles they shall go. Answere. Double the nomber of his mylles that goeth eche day an egall nomber of myles And of the nomber dowble take awaye one vnyte, and the remanant shall shewe you what daye they shall mete eyther o­ther

¶The rule and questyon of the women that bare apples to the market.

THre women bare apples well & ho­nestly trymmed to the market, of whome the one bare 50 the other 30, and the thyrde 10, theyr housboūdes were brethern and gaue cōmaūdement to them that they shold make as good market one as an other, that is, that they sell all after one pryce, & that the one brynge as moch money home as the other. I demaunde howe that maye be done. Answer. It is [Page] possible. For fyrste there cometh a mar­chaunt to her that hath 50 appels: and sayth to her howe many for one peny, and she answered 7 and so she maketh 7 d̔. of her 50 appels and hath remaynynge one apple. The other solde after the same pryce. And she that had 30 apples solde hers for 4 d̔. and she had remaynyng 3 apples. The other that had 10 apels solde hers for 4 d̔. and she had remaynge 2 appels. And then came there anothe marchaunt that gaue 3 d̔. for an apple. And so eche one bare home 10 d̔ as ye se in this ensample. And thus may ye do of all other semblable.

❧The rule and questyon of the bagge

A Marchaunt hath a bagge that weygheth 19 ounces of thre mettalles, wherof 7 ounces ben of golde 8 of syluer, and 4 of copre. And he wyll take therout 5 ounces. I demaunde howe moche of golde, howe moch of syluer, and howe moche of copre is in these 5 ounces. Answere, ye shall multyplye the 5 for to [Page] knowe the golde by 7, for the syluer by [...] and for the coper by 4. And for to fynde the deuysor, ye shall sette all the multiplycators togyther, that ben 19, therfore de­uyde by 19. The answere is in this ensample 5 ounces.

• 7 Of golde 1 ounce an halfe 8 pens 5 grayne. Resteth 1 pens.
• 8 Of syluer 2 ounces, 2 pens and halfe, 1 halfe grayne. Resteth 2 pens.
• 4 Of coper 1 ounce 1 peny 6 graynes. Resteth 6.

¶Now set the remeynaūt togyther and dyuyde it by the dyuysor cōmune, that is 19. And it is 1 halfe grayne.

¶The rule and question of the bell.

IN a chyrche is made a bell, and there in is put 30 pounde of golde 50 li. of syluer, 100 of tynne, and 102 of coper. Whan the bell is made there remayneth 40 pound in one pyece, that they wyll sell I demaunde how mych is there of golde howe mych of syluer, how moch of tynne, [Page] and how moche of copre. Answere. ye shall do as aboue is sayd of the bagge for ye shall multiply 40 eche one by hymselfe, and dyuyde by 282.

¶The rule and questyon to chaunge golde into syluer.

A Marchaunt hath 100 trancz in gold and he goth vnto a chaūger & sayth, I haue 100 francz in pyeces of golde I wolde haue the money therof in small pyces, that is to wytte, of 2 pens, of 3 pens of 4 pens, of 5 pens, of 6 pens, of 8 pens, 4 of 10 pens, & I wolde haue as many pyecz of one as of an other. I demaunde howe many pyeces of euery money oughte the chaunger to gyue hym. Answere, ye must set togither all these nombers 2, 3, 4, 5, 6, 8, and 10, that ben 2 the deuysor commune and then ye must make of the francz pens that is 24000 pens, whiche ye shal deuyde by 32 and there bē 750 pyeces of eche mo­ney, and thus ye may do of all other sem­blable.

¶The rule and question of cloth of dyuers colours

I haue a pyece of clothe wherof the thyrde part is whyte, the fourth part blacke, and 8 elles of graye. I de­maunde howe moch hath it of lenghte. [Page] Answer. Set 12, for in 12 ye shall fynde one thyrde and one fourth, the thyrde and the fourth of 12 is 7, and there remayneth 5 / therfore forme the rule of thre, yf 5, be co­men of 12, of how moche shall come 58 multyply 12, by 3 that ir 96, & dyuyde by 5, and therof cometh 19 elles and 1 fyfte, therfore ye maye answere that the pyece of clothe hath of lenght 19 elles and one fyfte.

¶The rule and questyon of spyceryes.

¶A bourgesse sayd vnto h [...]s seruaunte holde these 13 frances, and go and bye the peper that costeth 15 sꝪ. the pounde, and sugre that costeth 18 sꝪ. the pounde, & of fyne spyces that costeth 9 sꝪ. the pounde, and gynger that costeth 13 sꝪ. the pounde, and cloues that costeth 10 sꝪ. the pounde, and brynge me as many poūdes of one as of another. I demaunde how many poūdes oughte the apotycarye to gyue hym for 13 francz. Answer. ye shal set al the pryces togyther 15, 18, 9, 13 and 10, that ben 65 which shall be the dyuysor, & then ye shall make [Page] the francz in shyllynges, that is 260 shyl­lynges. And than ye shall deuyde by 65, & therof cometh 4 pound / therfore ye maye answer, that he ought to gyue hym 4 poū des of all these spyceryes.

¶The rule and question of the egges.

A yonge mayden bayreth egges to the market for to sell and hyr meteth a yonge man that wold play with hyr in so mich that he ouerthroweth & breketh the egges euery one, & wyl not paye for them. The mayde doeth hym to be called afore the iudge. The iudge condempneth hym to pay for y^e egges, but the iudge knoweth not howe manye egges there were. An that he demaūdeth of the mayde, she answereth y^t she is but yonge, and can not well compte, but she and hyr moder had ordeyned and dysposed thē by 2 and 2 & there remayned 1 egge. Than by 3 and 3 & there remayned 1 / than by 4 and 4 and there re­mayned 1 / thā by 5 & 5, & there remayned 1 / than by 6 and 6 & there remayned 1, and at the last by 7 and 7 and there remayned [Page] none / I demaūde how many egges there were. Answer. 721. And for to proue it, multyply the nōbers one by another in sayēge 2 tymes 3 ben 6, 4 times 6 ben 24, 5 tymes 24 ben 120, 6 tymes 120 ben 720, and set therto 1 that remayned always & than they ben 721 that which ye shal deuyde by 7 / & there remayneth nothyng / and so she hadde 721 egges. And fater this ensample maye the iudge iuge the yonge man to pay.

¶The rule and question of money for­gotten with a chaungeour.

AN aduocate hath gyuen to a chaun, geour money, & hath forgotten howe moch. For to know howe moche and for to haue all his money, he fyndeth subtyltye that ensueth, he sayeth to one of his sonnes, of whom he hath many go vnto suche a chaūgeour & bryng me a frāce and the tenth part of the money that I delyuered hym, and so was it done And an other tyme he sayd vnto another sone, go vnto the chaūgeour & brynge me 2 frāces and the tenth part of the remaynant, and so he sayd vnto all, but vnto y^e last he said [Page] vnto the chaungeour, and brynge me all the remaynant of the money, and so was it done, and as mouche brought y^e one as the other. I demaunde how moche mony he hadde, how many sonnes, & how moch money eche one of them brought. Answer For this thre questyons pose the nomber that they all brought, that is to wytte, the tenth ben 10, and of 10 take one and there do remayne 9, therfore ye may say that he had 9 sonnes, and eche one brought 9 sꝪ. And for to knowe how moche he had gy­uen to the chaungeour, ye must multypyl 9 by hym selfe, and it is 81. Therfore he had delyuered 81 frances to the chaunge­our. For to make the proue lay 81 and take vp for the fyrst sone 1 and the tenth parte of the remaynant, and in lyke maner ye muste do of all other.

¶The rule and question, of tyme. &c.

¶A mā sayth yf I hadde as moch more of tyme as I haue, and y^e halfe, the thyrd and the fourth of my time that I haue set to. I sholde haue of yeres 50, I demaund [Page] what age he hath. Answere. Laye 12, for in 12 ye fynde an halfe, a thyrd, and a fourth And then set there ones as moch, & that ben 24, than set therto 1 halfe, 1 thyrde, & [...] fourth of 12, and they ben 37, and thenne fourme thy question. yf 37 be comen of 12 of howe moch shall come 50. Multyply 12 by 50 and dyuyde by 37, and ye shall fynde that he hath 16 yeres 78 dayes and a halfe 10 houres resteth 2.

¶The rule and questyon for to deuyde dystrybucyons.

IN a churche bell 12 channons 9 prestes and 6 clerkes they haue to dyuyde a dystrybucyon of 400 frnces, werof the chanons haue 3, the prestes 2 and the clerkes [...], I demaunde howe moche shall haue the chanons, how moche the prestes and howe moche the clerkes. Answere. Multyplye one nomber by an other in sayenge 3 tymes 12 ben 36 that is the mul­typlycatour for the chanous, 2 tymes 6 ben 18, the multyplycatour for the prestes, 1 tyme 6 ben 6, the multyplycatour for the clerkes. Howe moche eche one oughte [Page] to haue ye may se in the ensample by the deuysor. Set togyder al the multyplyca­tours & they ben 60, the deuysor comune.

¶The rule and question of the speyre.

A Speyr is the halfe and the thyrde part within y^e water, and 9 fote with oute. I demaunde howe myche of lenghte hath the spere. Answere. Set 6, for in 6 is founde a halfe and a thyrde the halfe and the thyrde of 6 ben 5, and there remaneth 1, forme the rule of three yf 1 be comen of 6, of howe many shall come 9 multiplye 6 by 9, and they ben 54, deuyde them by 1 and they ben 54, therfore ye may answere that y^e spere hath 54 fote of lēgth, the halfe is 27, & the thyrd is 18, and there bē 45 fote within the water, and 9 without that is 54. And so mayr ye do of all other semblable, as of a toure.

¶The rule and question of two mē that went that one agaynst that other.

TWo men begin to go and take theyr iourney that one agaynst that other vpō one daye and in one houre. For that one that goeth fro Parys, to Londō and goeth euery daye 7 myles, that other goeth frome Lyon to Parys, and goethe eche daye 9 myles, and from Lyon vnto Parys ben 80 myles. I demaunde howe lōge tyme shal it be or they mete. Answer Set togyder the myles that they go in one daye, y^e is to wyt, 7 and 9 ben 16, forme now the rule of 16 come of 1 daye, of howe moche shall come 80 that they haue to go, multyplye 80 by 1 & it is 80 they whiche he may deuyde by 16 & therof cometh 5, therfore in 5 days they mete. The proue is, for he y^e from Parys to Lyō goeth in 5 dayes goeth 35 myles / & that other 45 the which ben 80 myles.

¶The rule and questyon of a catte.

THere is a catte at the fote of a tre the lēght of 300 fote, this catte goeth vp [Page] warde eche daye 17 fote, and descendeth the nyght 12 fote. I demaunde in howe lōge tyme shall she be at y^e toppe, Answer. Take vp and abate the nyght of the day, that is 12 of 17 and ther remayneth 5, ther­fo [...]e the catte mounteth eche daye 5 fote / deuyde now 300 by 5 and therof cometh 60 dayes then she shall be at the toppe. And thus ye maye do of all other semblable. For of this rule ye may make 4 questy­ons, as it appereth in the practyse therof.

¶The rule and questyon of 20 scolers.

IF [...]0 scolers owe vnto theyr host 5 d̔. tourneyes, how oughte they to paye, so that eche one pay his duty & gyue the money of his purse. How moche shall eche one paye. Answere. Eche one shall pay 1 peny Parys, and the hoste shall re­thurne vnto him agayne 1 peny tournoys and so eche one shall paye the 4 parte of a tournoys.

¶The rule and question of pylgrymes.

¶Twenty pylgrymes, that is to wytte, [Page] men, women, and lyttell chylderne, haue spended in drinke 20 pens, wherof the men paye 3, pens, the women 2 pens, and the lytell chyldren halfe pens. I demaūde how many men, & how many women, and how many chyldren be there, for to pay this 20 pens, so that there be 20 persons. Answer. There shall be one man. 5 women, and 14 chyldren.

¶The rule & questō of a chauntour.

¶A chauntour hathe eche daye of rente fro the courte of the prynce 12 sꝪ. the which is payed by knyghtes, damoyselles, and squyres of whome the knyghtes paye 2 sꝪ the damoysels 6 pens, and the squyers 3 pens. I demaunde how many knyghtes how many damoysels, & how many esquyres ought there to be, to paye this 12 sꝪ. so that there be 12 persones. Answere. There must be 5 knyghtes, 1 damoysell, and 6 esquyres.

¶The rule and questyon for to dyuyne,

¶If ye wyll cause your felowe to by­leue that ye shall dyuyne howe many pyeces [Page] of syluer he hath in his ryght hande say vnto him that he put as many pyeces in that one hāde as in that other. And thā that he take fyue from y^e lyft hande to the ryghte hande and than that he put forth of the ryght hande into the lyfe hand as many pieces as he hath remayning in the lyfte hand. And there shall remayne 10 in the ryght hande.

¶The rule and questyon of thre saynctes.

A Holy hermyte is ētred within a churche wherin there ben thre sayntes: y^t is to wyt / saynt Peter, saynt Paule, and saint Francoys / this hemite cometh fyrst to saynt Peter and sayth to him in a maner of his oraysō, I pray y^e that it pleas the to double me the great blances that I haue in my purse, and I shall gyue the 6, and so was it done. Thā came he te saynt Paule & sayde to hym, please it the to double me y^e great blances that I haue in my purse and I shall gyue the 6 & so was it done. Then came he to saynt Francoys & sayde, yf it wold [...] please to double me the [Page] great blances that I haue in my purse I shall gyue the 6, and so was it done, and nothynge had he remaynyng. I demaūd howe many greate blances had he in his purse. Answe. He had 5 and 1 fourth. And for to know it double them and they ben 10 and an halfe, and then ye must giue 6 to saynt Peier, and there remayneth 4 & an halfe, double them. & they ben 9 And then gyueth he 6 to saynt Paule, & then there remayneth 3 double them and there ben 6 and that 6 gyueth he to saynt Francoys, and so he hat nothynge remaynynge.

¶Here folowe dyuers other proper rules and questyons.

A Lorde hyreth a seruaunt, the whiche he sholde gyue euery yere [...]0 nobles and a gowne, and the same seruaunt dwelleth 7 monethes with hym, and then they varye in so moch that his lorde gaue hym lycence to go his way. And sayth, go thy wayes out of my house and take thy gowne with the, and then I am nothyng in thy dette. Now I demaund what was the gowne worth, wyll ye know that, then [Page] marke how many monethes 7 is lesse than a yere, that is 5 monethes lesse. And had y^e seruaūt taryed so long yet by his mayster than shold he haue had the gowne & 10 nobles. Therfore saye thus 5 monethes gy­ueth 10 nobles, what giueth 7. Make it after y^e rule of thre / & it cometh 14 nobles.

¶Of thre felowes or yonge men.

¶Thre felowes play togyther y^e one to wynne the others money. For the one had more money than the other. And the fyrst casteth, y^t the one of them thre leseth iust so mych money as y^e other two hadde. Than casteth the second and leseth also as mych as the other two hadde Then casteth the thyrde and leseth also iust as mych as the other two had. And than was the money iust deuyded, & had eche lyke moch. Now I demaunde how moche had eche or they began to playe, & how moche money that eche had whan they played. Wyl ye know that, then marke how many persons dyde playe, & adde 1 to them, as here adde 1 to 3 maketh 4. So many nobles had y^e fyrst. Now dowble 4 cometh 8, & subtra 1 from 8 [Page] rest 7: so many nobles had y^e seconde. Thē double 7 cometh 14, therof subtra 3 reste 13, so many nobles had the thyrd.

An other question.

A Man byeth 46 pounde of saffron for 30 pounde, what shall cost 63 poūdes of saffron. wyll ye knowe that, then multyply the 30 poundes with the 63 poundes of saffron, cometh 1890. Now deuyde them with 46 cometh 41 poundes and 4/49 parte of a pounde to paye for the 63 poundes of saffron. Now wyll ye knowe howe many shyllynges that 4/46 parte of a li. is. than multyply 4 by 20, for 20 sꝪ maketh a li. cometh 80 sꝪ. deuyde them with 46 com­meth 1 sz, and 4/64 part of a sꝪ. Now wyll ye knowe how many pens that 34/64 parte of a shyllynge is, thē multiply 34 with 12 12 pens maketh a sz, cometh 408. Dyuyde them wyth 46 cometh 8 pens and 40/46 part of a pen̄y. Now wyl ye knowe how many farthynges that [...]/4 [...] parte of a peny is, then multypplye 40 with 4, for 4 farthyn­ges [Page] maketh a peny, comet 160 farthyn­ges. Now deuyde them with 49, cometh 3 farthynges and 22/46 parte of a farthynge, Thus done ye shall fynde that 63 li. of, saffron coste 41 li. 1 sꝪ. 8 farthynges and 22/46 parte of a farthynge.

¶Item a 165 poundes of alome coste 2 poundes 5 shyllynges 6 pens 9 fa [...]thyn­ges: what shal cost 22 poundes of alome. If ye wyll soyle this question, than make of your poundes shyllynges & adde ther­to the odde 5 shyllynges: commeth 45 sꝪ. Then make of the 45 sꝪ. penes, and adde 6 pens, commeth 546 pens, than make of your pens farthynges, and adde ther­to the 9 odde farthynges, cometh 2193 far­thynges. Now multyply the farthynges with 22 cometh 48246 farthinges. Now deuyde them with 165 cometh 592 and 66/165 part of a farthynge, for so many farthynges shall coste 22 li. of alome. Nowe wyll ye knowe how many pens that the forewry ten farthynges make, then dyuyde them [Page] with 4, for 4 farthynges make a peny. Then wyll ye knowe how many shyllyn­ges that they make, then deuide the pens w^t 12, for 12 pens maketh a sꝪ. Thus done ye shall fynde that 22 pound of alome cost [...] sꝪ. 3. d̔. 1 farthyng, and it is done.

¶An other questyon.

¶A marchaunt hath bought a bagge of peper, I saye not how heuy, but whan he gyueth for a pound of peper 12 pens, then remayneth hym yet 37 d̔. And when that he gyueth for a pounde of peper 15 pens, then he lacketh 44 pens to paye for the peper. Now I demaūd how heuy the bagge of peper was, and how moch money that the marchaunt had. For to know this & suche other lyke questyon, ye shall take and subtra 12 from 15 and there resteth 3, whiche 3 shalbe your deuysor. Then shall ye adde 44 and 37 togyther / and that ma­keth 81. Then muste ye deuyde 81 with 2, & therof cometh 27, so many pounde wayeth the bagge of peper. Now wyll ye knowe how moche money the marchaunt hadde, [Page] then must ye multiplye 12 with 27. and adde 37 therto, or multyply 15 with 27 and subtra 44, comith 361, so many pens hadde the marchaunt.

An other questyon.

A Dronkart drynketh a barell of bere in the space of 14 days, and when his wyfe drinketh w^t him than they drincke it out within 10 daies. Now I demaū de in what space that his wyfe shold drincke that barell of bere alone. For to soyle this questyon & suche other lyke, ye shall fyrst subtra the lee [...]t drynker from y^e more that is 10 from 14 and ther remayneth 4, & that is your deuisor. Now saye 4 gyueth 10 what gyueth 14 Make it after the gol­den rule, and ye shall fynde that she shold drynke it in 35 dayes.

¶Of addycyon.

ADdycyon is none other thyng but to set togyther 2 or 3 nombers and to make of them a totall somme, as in the ensample folowynge.

¶There is a man whyche owyth 20 li. 18 pounde, 100 pounde, 50 pounde, and 69 pounde. Now yf ye wyll knowe howe. mych y^t al these sommes maketh togyther Then for the first somme ye must lay two countres besyde the seconnde lyer, for the two stande for 20 / that is for the fyrste somme. Now for the seconde somme laye 1 counter besyde the seconde lyer, for that [Page] is 10 / and lay 1 counter betwyxt the ne­thermost & the seconde lyer, for that 1 stan­deth for 5 / and then lay 3 counters be­syde the nethermost lyer / and they all to­gether make 18. Now for the thyrde som ye shall laye 1 counter besyde the thyrde lyer / for that is an 100. For the fourthe somme laye 1 counter besyde the thyrde and the seconde lyer, that is 50. Now for the fyfte somme lay 1 counter betwyxte the thyrde and the seconde, and 1 besyde the seconde lyer, and 1 bewene y^e seconde and the nedermost, and 4 besyde the ne, thermost lyer, and that maketh togyther 69 / and in so doynge ye shall fynde that all the forewryten sommes make togither 247 as ye shall se in the fygure folowyng And euermore for a generall rule remem­bre your places, for euery counter that lyeth besyde the fyrst lyer standeth but for 1, in the second place euery counter stan­deth fof 10, in y^e thyrde place for 100 as is afore rehersed.

¶Wyll ye proue whither ye haue added well or not, than subtra all your sommes one after an other. And in lyke wise as ye do with this ensample so ye shall do with all other of addicion,

❧Of subraction.

SVbtractiō is, if ye wyll withdrawe any summe from an other summe, ye must knowe two numbers, that is to wytte, the nomber that ye wyl with­drawe, and the nomber where fro ye wyll withdrawe. An ensāple. There is a man that oweth you 9756 poundes, and there vppon he hath payed you 5989 poundes, Now yf ye wyl knowe what there resteth then set downe your sūme that he oughte you, & therof withdrw the sūme that he hath payed you, and that y^e remayneth is the som that he doth yet owe you, as ye more playnly maye se it in the en­samples hereafter fo­lowynge.

And when ye haue set your deth / y^t is to say 975 [...] poun̄des vnder this maner as a fore shewed. Then yf ye wyll knowe the rest, thē take therof that ye haue payed as 5989 poūdes. Now for to do this / ye shall fyrst take vp the coūter that lieth betwene the fourth & fyft lyer for that is 5000. Thē take vp one of the coūters which lyeth besyde the fourth lyer, & that is a thousād / & ye shold take away but 900, therfore ye must lay downe 1 coūter agayne besyde y^e thyrde lyer, that is a hōdreth. Thē take vp one of the counters that lyeth besyde the thyrde lyer, whyche is a hondreth / and ye sholde take vp but 80, therfore ye muste laye 2 counters besyde the seconde lyer / that is 20 and 80, that ye haue take vp maketh 100, then take vp one of the coun­ters that lyeth besyde the seconde lyer / y^t is 10 / and ye sholde haue take awaye but 9, therfore ye must laye one counter be­syde the nethermost lyer, that is 1 and the 9 that ye haue subtrahed or take vp, ma­keth 10, and there remayneth 3767 pounde det, and stande thus.

Wyll ye proue whyther ye haue subtrahed well or not, then adde ther­to that ye haue payed, and yf the somme come then so great as it was afore, then is your subtraction true, els not.

Multiplication.

MVltiplycation is nothynge e [...]les but to multiplye one nomber by an other, as thus, to know what is 6 tymes 9 or 6 tymes 12 and suche lyke. And in multiplycation ye must consyder two nombers, that is to wyt, the nomber that ye wyll multyply, and the nomber wherby ye wyll multiply, and ye muste worke in multiplycatyon after this ma­ner: Fyrste ye shall laye downe the lesser nomber, which is 6, and this 6 is the nō ­ber that shall be multyplyed, and the 9 is the nomber that ye shall mutiply withall And ye shall laye the nomber that shal be multiplied, at the right syde of your lyers & when ye worke your multiplycation, ye [Page] shall laye them at the lyfte syde, as in this ensample here after folowyng shall more playnely appere.

¶Fyrst ye muste laye downe the lesser nomber, whych is 4, as ī this ensample, as ye se them layde here on the ryght hande of the lyers. And when that ye ha­ue thus done, ye muste take vp one coun­ter and laye 9 for it on the other syde of the markes, that is to wyt / at the lyfte syde. And after that take vp another coū ­ter / and laye also 9 for it / and so forthe for euery counter that ye take vp ye must laye 9 for it at the other syde. And when that ye haue so wrought your worke, it wyll come iuste to 36, as ye se the coun­ters before layde on the lyfte hande of the lygnes.

¶And yf ye wyll multiplplye by greater numbers as thus / to know what is 24 tymes 14. Fyrste laye 14 on the ryghte hand of your lyers or markers, as this ensample folowynge sheweth.

place nexte aboue your fynger / and reke [...] euery chone of them for 2 whiche maketh iuste 6, then 10 and 6 maketh 16 / as the fygure before shewed / when ye haue so done, then take away your fynger, and for euery one of the 3 counters that lyeth in the fyrste place on the ryght syde lay 16 on the lyft syde / and then take them of the ryghte syde away / and ye shall se that the nomber shall iuste come to 128, as the en­sample before shewed. And thys wyse ye muste reken all counters that lyeth in the spaces yf the multplycacyon shalbe truely made.

❧Of dyuysyon.

DYuysyon is to deuyde a somme thorough an other somme: and in this dyuysyon must beknowen two nombers, that is the nober that ye wyll de­uyde, and the nober wherby ye wyll de­uyde it, as to know how many tymes ye [Page] maye haue a small nomber out of a great as by ensample, yf ye wyl deuyde 336, by 14 as in this ensample hereafter. Fyrste laye 336 on the ryght hand of your lyers, & then set your fynger at y^e hyest place where any coūter lyeth, for as I shewed you before, y^e damneth all the other places beneth, so y^e then there as your fynger is, is the fyrste place. And then loke ye may take 14 frō that place, whiche ye can nat do, for euery counter stādeth but for one, bycause your fynger is there, therfore ye must remoue your fynger to the nexte place benethe where the other 3 counters lye / and then loke yf ye maye take 14 from that place whyche ye maye do ryght well / for these 3 counters at youre fynger standeth but for, 3 and the other 3 counters aboue standeth for 30, and then se howe many tymes 14 ye may haue out of 33 and so many counters ye must laye on the other syde iust agaynste your fynger / that is to saye, ye maye haue 28 out of 33, that is two tymes 14 out of 33 / and therefore ye must take vp 28 and laye 2 counters on the

other syde agaynste your fyn­ger, & thā ye can haue 14 no more, than ye muste remeue your fynger to the next place beneth, & then rekē that place at your fynger to be the fyrst place, as ye dyd before: & then loke how many tymes ye maye haue 14 from that place, which ye can not, for that counter at your finger standeth but for 1, & the other in y^e space aboue standeth but for 10 whiche is in all but 11, therfore ye muste remoue youre fynger to the nexte place beneth / and then ye shal se that that nomber is 56 out of the whyche ye may well take out 4 tymes 14 whyche maketh iuste 56 therfore ye must take vp 56 and lay 4 counters on the other syde agaynste your fynger / and then take away your fynger, and ye shall se that that nomber that ye haue layde on the lyfte syde of your markes cometh iust to 24 as the ensample before shewed, and [Page] then ye haue your question soyled; for yf ye deuyde 336 by 14 it cometh iust to 24 for 24 tymes 14 maketh iuste 336, as I haue shewed you before in the rule of multyplycation. And lykewyse as ye haue deuyded this nomber, ye may do with all other nō ­bers. And yf ye wyll proue whither ye ha­ue well deuyded or not, take the nomber that cometh of your diuisyon, and multyplye it with the smal nomber that is your deuysor, and adde that remayneth therto yf there be any, and than it wyll come iust to the great nomber that was the nomber to be deuided. And lykewyse yf ye wyll prone whyther ye haue truely multyplyed or no, take y^e great nomber that cometh of your multiplyca­tiō, and deuyde hym by the nomber that is to be multyplyed, and it wyl come iust to the thyrde nom­ber that was your multyplyer,

¶yf ye desyre to knowe how many gro­tes ben in 79992 pens. Fyrst ye shall set downe your pens as ye se in this fygure afore, and ye shall deuyde them with 4 for 4 d̔ maketh a grote. Now to the operacyon therof, whan that ye haue set downe youre pens, as the fygure afore shewed / than set your fynger at the hyest counter. and se yf ye may haue 4 from that place / which of a suretie ye can not, for there ly­eth but one, and it standeth but for one, by cause you fynger standeth there therfore ye shal remeue your fynger and sette it a­gaynst the fyfte lyer, and se yf ye maye take awaye 4 the whiche ye may do, for there lyeth 7. Now your fynger standeth agaynste the fyfte lyer, therfore ye shall take vp the counter that lyeth in the next space aboue your fynger, for that counter is 5, and ye sholde take vp but 4 therfore ye shall take it vp and lye it besyde the fyfte lyer of the ryght syde, and lay 1 on the lyft syde also beside the fyfte lyer, then se yf ye can haue 4 any more from y^e place y^t which ye can not, therfore remeue your [Page] finger and set it against y^e fourth lier, and then se how many tymes 4, ye maye haue out of 39 ye may haue 9 tymes 4, and for this 9 tymes 4 ye shall lay 1 counter in the space betwene the fourth and fyfte lyers, and 4 besyde the fourth lyer, & that make 9 And as ye haue done with these so shall ye do with all other folowyng, & whan ye haue fynished your worke ye shal fynde y^e 79992 pens make 19908 gretes as ye may se playnly here in this fy­gure.

¶The proue.

¶If ye wyl proue whyther ye haue deuyded well or not, then multyply the grotes with 4, for 4 pens maketh a grote, and yf the some come to stande as it dyde afore then ye haue deuyded well.

¶Itē when ye deuyde ony somme with d̔. yf there remayneth any thing it is pens And yf ye deuyde by shyllynges, yf there [Page] remayneth any thynge it is shyllynges. And as ye haue done with these forewry­ten examples so maye ye do with al other.

¶The golden rule.

¶Regula aurea is called y^e golden rule for lyke as golde passeth all other metall so this rule passeth all other rules in aw­grym. And to the operacyon of this rule muste alwayes be noted thre thynges or thre nombers, of the whiche two of them must be lyke of names and of kynde, that is to wytte, the fyrste and the thyrde nomber, & alwayes ye shall multyply the se­cond nomber with the thyrde, & that that cometh of the multiplycation is the nom­ber to be to deuyded by the fyrst nomber that is general dyuysor / and the quocient of the dyuysor, sheweth a nomber of solu­cyon of name and kynde of the mydle nō bers, as in these ensamples folowynge shall appere.

¶If a man byeth 40 egges for 20 pens how many for 12 pens, yf ye wyll soyle this question, ye must mutyplye the seconde and the thyrde nomber togyther / and [Page] the producte or the somme that commeth of that multyplycation, ye shall dyuyde by y^e fyrste nomber, lyke as here is shewed in this ensample, when ye bye 40 egges for 20 pens. what shal one pay for 12 egges ye shall multyply 20 with 12, commeth 240, the whiche ye shal deuyde with 40 cometh 6 pens, and so moche shall ye paye for the 12 egges. And thus ye maye do with al o­ther suche questyons.

❧An other qustyon.

¶Itē a 100 apples cost 12 pens, what shal one paye for 87, ye shall multyply 12, with 87, cometh [...]044, the whiche ye shall deuide with a 100 cometh 10 pens 44/10 parte of a d̔. for the 87 appels. Wyll ye knowe how many farthynges that the 44 hondreth parte of a peny is wdrth, then multiplie 44 with as many farthynges as a peny is worth, that is 4 cometh 176 the whiche ye shall deuyde with 100 cometh 1 farthynge and 67/100 parte of a farthinge.

❧Item 165 pounde of waxe cost 2 li. 5. sꝪ. 6. d̔. 9 mytes. what shall coste 22 l. For to soyle this question and suche other lyker, [Page] fyrst ye muste make of poundes sz, & adde therto the odde 5 sꝪ. the whiche stande in this question, and the come togyther to 45 sꝪ. then make of the sꝪ. pens, and adde thereto the odde 6 pens that standeth in y^e questyon cometh 546 d̔. than make of the pens mytes, and 24 mytes is a peny, and therto adde the odde 9 mytes, that stan­deth in the question commeth togyther to 23113 mytes, and that is the totall som̄e of all the li. sꝪ. d, & mytes togyther Nowe make it after the rule and say 165 pounde of wax cost 13113 mytes, what shall coste [...]2 pounde, fyrste multiplye the myldel­most and y^e laste togyther, that is ye shall multyplye the mytes with the last, that is with 22, and it shall come to 388486, deuyde them with 165 and it shall come to 1748 my­tes and 6/165 parte of a myte, so many my­tes shal the 22 pounde of waxe coste. Now wyll ye knowe how many pens that the forewrytē mytes make: then deuyde them with 24, for 24 mytes maketh 2 peny Then wyll ye know how many sꝪ. that the pens make, then deuyde them with 12, for 12 d̔. [Page] maketh a sꝪ. And thus doing ye shal fynd that the 22 li. of wax shal cost 6 sꝪ. 20 mytes & 66/165 patt of 1 myte and it is done.

¶Item when there standeth 1 in the fyrst place, As 1 goos cost 3 d̔. what shall cost 28. ye shal multyply the myddels [...]e with the last in sayeng 3 tymes 28 is 84 so many pēs cost 28 gese, and it is fynyshed.

¶Item in the cōtrary as whan that 1 cometh in the latter ende, as here in this en­sample 20 capons cost 23 pens / what shall cost 1 capon. For to soyle this question ye shall deuide y^e myddelmost with the first that is 23 with 20 cometh 1 d̔. and 1/20 part of a peny, that is 0 mytes & ⅘ part of 1 myte for 1 capon,

☞Item 17 elles and ½ cost 14 nobles and 1/ part of a noble, what shall cost 32 elles and 2/ [...] part. For to soyle this questyō and such lyke / fyrst ye must breke the first and last broken togyther crossewyse in sayeng 1 times 4 is 4, set the 4 by 17 elles Thē say 3 tymes 2 is 6 / set y^e 6 by the 22 elles. Then multyplye bothe y^e nombers of your frac­tyons [Page] togyther / in sayenge 4 tymes 2 is 8 the whiche ye shall set vnder 4 and 6 / then it standeth thus: as in the rule of thre / yf 17 elles & 4/8 parte of an elle cost 14 nobles & ⅓ part of a noble / what shal cost 32 elles and [...]/ [...] part of an elle. Fyrst mulyply the fyrste hole nomber with the nethermost of his broken that is 17 with 8 cometh 136 & therto ye shall adde y^t 4 that standeth a­boue 8 comeeh 140, the whiche ye shal multyply with the 3 that standeth vnder the seconde broken com [...]th 430 that is your deuysor, then multiply 14 with the 3 that stā ­deth vnder 1 and adde that 1 therto cometh 43 that is your multiplicatour, then multyply 32 with the nethermost figure of the broken, that is with 8 and adde therto the same 6 that standeth aboue the same 8 co­meth 262. Now set it in the rule of thre in sayenge 420 elles cost 43 nobles, what shal cost 262, multiply the secōd with the thyrd and then deuyde that that cometh of that multiplycation with the fyrste, and ye shall fynde that the 32 elles of cloth cost 26 [Page] nobbles 16 d̔. 34 mytes and 150/420 parte of a myte.

¶Item whan that there standeth at the begynnyge a hole nomber with a broken and in the seconde and thyrde place no broken, as here. 36 elles and 1/1 coste [...] li. what shall coste 16 elles. For to soyle this question, ye must multyply the fyrst hole nomber with the vndermoste fygure of his broken, that is 36 with 2, and adde the 1 thereto that standeth aboueth 2 com­mynge 73, and that is youre deuysor, then multiplye 8 li. also with the vndermoste fygure of the broken, that is to wyth with 2, and it cometh to 16, then multyplye 16 withe 16 cometh to 259 the whiche ye shal deuyde with 37, and it wyll shewe you that there shall be to paye for the foresayd clothe 3 pounde 10 shyllynges 1 peny 17 mytes, and 2 [...]/73 parte of a myte.

¶Item when ther standeth in the fyrste nor in the seconde no broken nomber, but in the latter ende a hole nomber with a broken, as here 1 [...] ounces of grayne coste [Page] 14 sꝪ. what for 9 ounces of grayne and one thyrde. For to knowe this ye shall multiplye the fyrst 14 with 3 that is your deuysor, then multiplye 9 with 3 and adde that 1 therto y^e standeth aboue 3 cometh 28, then set it thus 4 [...] gyueth 14 what gyueth 18, make if forth after the rule and ye shal fynde that there is to paye 9 shyllynges 4 pens for the 9 ounces of grayne and one thyrde parte.

¶Item when that ye fynde neythere at the beginnynge nor at the latter ende no b [...]oken nomber, but in the myddes a hole with a broken as here. A man bought 48 shepe for 64 crones and [...]/4 what shall coste 18 shepe, For to soyle this questyon ye muste multyplye 48 with 4 cometh 192 that is your deuysor, then multyplye 64 with 4, and adde therto the 2 that stand­deth aboue 4, cometh 258, the whiche ye shall multyplye with 18 cometh 494 [...] the which ye shall dyuynde with 192. And thus ye shal fynde that ye muste paye for the 16 shepe 24 crones 4 stuners & 36 mytes

¶Item when ye fynde no broken at the begynnynge but in the seconde and thyrd one hole with a broken. As 7 elles for 6 pounde ¼ what shall cost 16 elles & [...]/ [...] For to soyle this question ye must multyplye the two vndermost borken nombers togyther, in sayenge 3 tymes 4 is 12, the whiche ye shall multiplye with 7 cometh 84 that is the dyuysor. Thē multyply eche with his broken cometh 25 and 46 y^e which ye shall multiplye one with the other, cometh 1225, the whiche ye shall deu [...]e with 84, and the solucyon shalbe to paye 14 li 12 sꝪ. 18 d̔.

¶when that ye fynde at the begynnynge and the myddes a hole with a broken, and at the latter end standyng a hole without a broken as 9 elles & [...]/4 for 5 pounde ⅝ what shall cost 15 elles multiplye 9 with 4 and adde therto 3 cometh 59, the whiche ye shall multiplye with 8 cometh 312, that is your deuysor then multyply 5 with 8 and adde therto the 5 that standeth aboue 8 cometh 45 the which ye shall multyply with [Page] 4 that standeth vnder y^e fyrste broken, cometh 180. Now set it in the rule of thre in sayeng 312 gyues 180 what gyueth 15, make i [...] after the rule and it cometh 8 li. 13 sꝪ. 22 mites and 1 [...]/59 of a myte.

☞The rule of company.

¶There be 3 marchauntes or compa­nyons the whiche laye togyther theyr money in marchaundyse, and eche to wynne after his inlayenge, wherof he fyrst layd in 170 crones. The seconde 60 crones The thyrde 40 crones and therwith they haue wonne 50 crones besyde al vncost Now I demaunde howe moche that eche shall haue after his inlayeng. Now for to soyle this question and all suche other rules of company, ye muste make of theyr mo­ney that they haue layd in a totall somme cometh 250 / now saye 250 gyueth 50 what gyueth 150 make it after the rule of ther and it cometh to the fyrste man 30 crones wynnynge. Nowe for knowe what the seconde hath wonne, ye shall saye 250 gyueth 50 what gyueth 60 make it after [Page] the rule, and ye shall fynde that the secō ­de hath wonne 12 crones, wyll ye knowe what the thyrde hath, then saye 250 giueth 50 what gyueth 40 make it after the rule, and ye shall fynde that the thyrde hath wonne 8 crones. And thus shal ye do with all other rules of company.

The rule of company with tyme.

¶Thre felowes doth marchaundyse togyther, wherof the fyrst layeth in 50 cro­nes for 4 monethes. The seconde 80 cro­nes for 2 monethes. The thyrde 100 cro­nes for 5 monethes, & withall this mony they haue wonne 6 crones besyde all vn­cost payde. Now I demaunde what eche hath wonne with his money, for to know this ye muste mulyplye eche mannes money with his tyme, that is for the fyrste 50 with 4 cometh 200, set the as thoughe he hath layde in so muche, for the seconde multyply 80 with 2 cometh 160 set it also as though he had layde in so moche, for the thyrde multyplye 10 [...] with 5 cometh 500, set that also as though [Page] he had layde in so moche. Nowe adde the 3 nombers togythere and then make it after the rule of companye and then shal ye fynde what eche hath wonne with his money.

The rule of baterynge.

¶Two marchaunte men wyll chaunge theyr wares togyther and the one hath a fyne blake clothe the whiche is 43 elles longe, and he wyll gyue the elle no lesse than 18 pens. The other marchaunt hath peper, and he wyll sell the pound no lesse than 13 d̔. Now I demaunde howe many pound of peper the fyrst marchaunt shall haue for his 43 elles of clothe. For to soyle this question ye shall say 13 gyueth 43 what gyueth 18, make it after the rule of 3 and ye shall fynde that the fyrste shall haue for his clothe 59 pounde of peper 8 ounces 12 engelsche and 4/13 parte of an en­glysche

☞Of a watte.

¶A wat ronneth in the fylde and ouer­ronneth in one mynute / there be 60 in an houre / 12 roddes of grounde. And a gra­hounde [Page] beynge her enemye / foloweth her and ouerronneth in one mynute 15 rod­des of grounde. But or the grahonde be­gan to ronne the hare had ronne 200 rod­des of londe. Now is to be demaunded in howe many minutes and how many rod­des of londe was the hare taken. For to soyle this question and suche lyke, ye shal subtra the lesse ronnynge out of the more that is 12 out of 15, and there remayneth 3 and therwith ye shall deuyde the space that the hare hath ronne afore or the gra­hounde began to ronne / that is 200 rod­des, And in so doynge ye shall fynde that the grahounde ouertoke the hare in the 66 mynutes and ⅔ partes of a mynute that is one houre and 6 mynues and ⅔ of a mynute, wyll ye knowe how many rod­des that the grahonde dyd ronne or that he toke the hare / then multiply 66 and 2/ [...] with 15 cometh 3000/ [...]3 the whyche ye shal deuyde with 3 cometh 1000 hole so many roddes dyd the hounde ronne or that he toke the hare.

¶The rule of two felowes.

¶Two felowes went togyther out of a towne, and the one goeth euery daye 12 myle, and the other goethe the fyrste daye but 1 myle, and the seconde daye 2 myles the thryde daye 3 myles / and so forth eue­re daye one myle more. Now I demaund in howe many days / and how many myle went he or that he ouertoke his felowe. For to soyle this questyon, ye shal double the myles of hym that wente euery daye lyke moche, that is 12 and 2 tymes 12 is 24, therof ye shall subra the one myle that the other goeth the fyrste day, and there resteth 23, vpon the same daye was the fyrste man ouertaken of his companyon? wyll ye knowe in howe many myle, then multiply 23 with 12 cometh 276 for so many myles went he or he ouertoke hym.

☞A man hath a golden crowne of 34 stu­uers, and a phylyppus gulden of 15 stu­uers, and a ducate of 28 stuuers, and with this money he goeth to the chaunger and wyll haue for it negenmannekens crow­nes of 9 mites, & of 3 mytes and of 2 mites [Page] and halfe mytes. Now I demaunde how moche that he shall receyue of eche for the foresayde golde and receyue of eche lyke moche. For to soyle this question and such lyke then make of all the greate money that he wyl chaunge mytes, for that is the leste coyne that he wyll haue / and cometh 9072 mytes, then loke howe many mytes that al the smal pens be worth that he wyl haue, that is 25. Nowe deuyde the greate somme of the mytes, that is to witte, 9072 withe 25 and ye shall fynde that he muste haue of eche 362 and 1 [...]/25 and it is done,

¶Of four carpenters.

¶Foure carpentes wyll make a house wherof the fyrst taketh vppō him to make it him selfe alone in a yere. The seconde wyll make it in two yeres, The thyrd wyl make it in 3 yeres. And the fourth in 4 yeres / Nowe I demaunde yf all these 4 wrought vpon that howse in what space wolde they 4 make that howse. wyll ye know that, then say the fyrst wolde make it in one yere that were 12 tymes in 12 yere The seconde in 2 yere, that were 6 tymes [Page] in 12 yere. The thyrd in thre yere that were 4 tymes in 12 yere. And the fourth in foure yere y^t were thre tymes in 12 yere. Nowe somme them all together that is 12, 6, 4, 3, cometh 25, therwith deuyde 12 cometh 12/25 prat of a yere Now yf ye wyll know how many dayes that it is, then multyplye 12 wthy 365 for so many dayes be in a yere / & that that cometh of that multyplycacy­on deuyde it by 25 cometh 175 dayes and 5/25 parte of a daye.

¶The rule of false posytyons, by the whiche all maner of difficult and harde questions maye easelye be dyssoluyd & fyrste of one salfe posytion.

NOw shal ye knowe how by false posytiōs or coniectures one or two ye shall fynde owt the very trewthe of that the which ye seake fore, and fyrste ye shall vnderstande how to fynd the trewth of a question proposed by one coniecture or posytyon.

When that any question is put furthe vnto you, too be assoylyd, of the whiche one part is knowyn and the other vnknowyn. Answere to that question by and by, with youre selfe at all auenture, and then cōsyder w^t your self whether ye haue made ryghte answere or no, yf not, loke what proportion is betwene your coniec­ture and that that folowyth of your con­iecture and the same proportiō is betwene the thing knowen and that y^e partaynyth vnto y^e selfe thyng beynge yet vnknowē. As by example ye shal more playnly per­ceaue,

¶A certayne way farynge man cōmyng by the way, founde so many crownes. that the seconde, the thyrde, & the fourth parte of them addyd togyther made 50, I demaunde what summe he founde. To make answere to this question by one posycyon, ymagyne some summe that hath these partes in it, that is to say: a seconde a thyrde, and a fourth part, and be it 12, whose seconde part or halfe is 6, thyrde parte 4, the fourth parte 3, whiche all addyd [Page] togyther 643, make 13. but the sum that he founde, the second, thyrde, and the fourth of it made 50. wherfore 12 is not the som he founde, therfore this position is false / & yet by this false shall ye come to the lyght of treweth, by the helpe of the rule of thre. For loke what proporcyon is betwene the seconde, the thyrde, and the fourthe part of 12 addyd togyther, the whiche maketh 13, and 12 whose partes therbe, the same proporcyon is betwene 50 whiche is the seconde, the thyrde, and the fourthe parte addyd togyther of the nomber vnknowen, & the same vnknowē nomber it selfe. Then saye thus with thy selfe: yf 13 whiche contayneth the fore­sayd partes in them, come of 12 of whome come 50, then set them thus 13, 12, 50, then by the rule of thre multyplye 50 by 12, and therof commeth 600 dyuyde the same by the fyrste nomber 13 and in the quotient thou shalt fynde 46 2/13 the whiche was y^e summe of the corwnes the which the man founde: of the whiche summe the halfe part is 23 1/13 the thyrd part is 15 1/1, the [Page] fourth parte 11 7/1 [...] the whiche partes ad­dyd togyther make iuste 50. Thus thou seest how that by one false posicion or cō iecture with the helpe of the rule of thre, this question is soone dissoluyd.

¶An other questyon.

¶Fynde me a number in the which 5 is ⅔ that is to say, two thyrd partes of hym Answer. ymagyne any number ye lyste 3 that hath thyrdes in it, as be it 6, then loke what is the thyrde parte of 6, that is 2. then two of this thyrde partes of 6 ma­kyth 4 wherfore this posicyon is false, yet by this false posicyon with the helpe of the rule of thre, thou shalt fynde out y^e trewth, after this maner. If 4 be the ⅔ parte of 6 to whome is 5 ⅔ partes serche by the rule of thre, thou shalt fynde it 5 ⅓

¶An other question.

¶What number is that in the which af­ter that the thyrde, the fourth, & the fyfte parte be deductyd out of it, there shall yet remayne 24? Answer. ymagen any nom­ber that hath a thyrde, a fourth, & a fyfte [Page] in hym. As for example: say it is 60, then subtrahe out of hym his third, his fourth and his fyfte part: and thou shalt fynde remayne but 13. Lo how muche thou hast myssed thou shuldest haue founde suche a nomber, in the which after the foresayd partes were subtrahyd shuld remayne 24 and here remayneth but 13, yet proue by the rule of thre, and thou shalt fynde the trewe numbe. If 13 remayne after the substraccyō of the aforesayde partes is 60 what number is that out of the which af­ter lyke substraccyō of his thyrde, fourth, & fyfth, parte shall remayne 24 proue by the rule & thou shalt fynde it 110 [...]/1 [...] whose thyrde part is 36 12/13 the fourth 27 10/1 [...] y^e fyfte 22 ⅔ which all added together make 86 20/ [...] the which deductyd out of 110 10/13 shall remayne 24. These and dyuerse o­ther questiōs before rehersyd by the same crafte one false posycyon maye soone be assoyled. Now wyll I shewe you howe to dyssolue all maner of questions how dyf­fyculte so euer they be by two false posy­cyons, [Page] For by one false posycyon ye shall [...] answere to all maner of questions, but two false posions, what euer que­styon it be / it may soone haue solucyon.

How to answere by two false posycyons.

INnumerable questyons do chaunce in numbres, the whiche though they can not be dyssoluyd by one posycion or coniecture yet shall it not mysse but be assoyled by two posicions: in the whiche maner ye must dilygētlye note howe farre aboue y^e trueth or vnder bothe posycyōs do fall: For by the obseruacyon of .ii. cō ­iectures how nere they be to the treuthe, & the dyfference of the errors whiche ensue of the posycyons, the verite commeth to lyght which may be done .iii. wayes: one waye by the rule of bothe more or bothe lesse. Another waye by the rule of the one more and the other lesse

¶when bothe posycions be more then the veryte or bothe lesse, then subduce the lesse errour out of the more errour, & that that remayneth shalbe the diuysor: then [Page] multyplye the fyrste errour by the seconde posycion, & the latter errour by the fyrste posycion, and then this .ii. numbers be­ynge multyplyed, deduct the lesse out of the more, and that that remayneh diuide it by the foresayde diuisor, and the quocient shall shewe the verite. Example ¶Thre marchaūtes diuided a 100 crow­nes so that the secōde shulde haue 3 crow­nes more then the fyrste, and the thyrde 4 mo thē the second: I demaunde now how many crownes eche of them receaued?

Answer. Fyrste make saynte Androwes crosse, as ye se hereafter, then coniecture what ye lyst, as for exāple, Say the fyrst had 33, and then muste the seconde haue 36, and the thyrde marchaunt 40, whiche sum gathered togyther maketh 109 but ye had but 100 to dyuyde, wherfore ye ha­ue myssed, and your posyciō redowndyth to more then the very sum by 9, whiche came of your fyrste posycion 33 wherfore set the fyrst posycion 33 at the vpper ende of the crosse on the left syde of the cosse, and the errour which hath ensued of that [Page] at the foote of the crosse on the same syde, as ye se in the example. And for bycause that this cōiecture came to more then the trueth, therfore set this letter M. in the space betwene the vpper ende of y^e crosse and the nether. And for as muche as in this fyrste coniecture ye haue erred thus much, coniecte agayne and spupose that the fyrste marchaunt had 31 then muste the seconde haue 34, the thyrde 38, all these collecte make 103, so that nowe ye haue erred agayne, your posycion beynge to much / so that your errour is 3, and for because that this seconde posyciō is more then the veryte as the fyrste was, set the posyciō 32 at the vpper ende of the crosse on the ryght syde, & the errour 3 the foote of the crosse on the same syde, & put this letter M. betwene the space to sygnyfey more. Bothe these pasycion then be more then the verite, wherfore accordynge to the rule fyrst subduce the lesse errour 3 at the fote of the ryghte syde of the crosse from the greater errour at the foote of y^e lefte syde of the same crosse, remayneth [...] [Page] to be set in the space betwene both y^e feete as ye se: which shalbe the dyuysor. Then accordynge to y^e rule, multiplye the fyrst posycyon whiche is 33 by the errour of y^e seconde posycyō which is 3, and therof cometh 99, then the secōde posiciō 31 by y^e er­rour 9 of the fyrste posycion, and therof cōmeth 279, then deducte the lesse sum 99 out of this more summe 279, remayneth 180, dyuyde this summe by the dyfference of the errours whiche is 6, and the quo­cyent shalbe 30 whiche is the trewe posy­cyon: For the fyrste man hauynge 30, the seconde must haue 33, and thyrde 37, whi­che all set togyther, make iust 100. Thus wonderfull craftly by these .ii. false posy­cyons y^e trewe & iust posycyō is brought to lyght.

☞An example when bothe posy­cyons come to lesse then the veryte.

WHen bothe posycyons come to lesse then is the veryte, the whiche is all one matter with the other, as ye shall perceyue by the same example agayne. As suppose ye had couiected that the fyrst had receued 27, thē most the seconde re­ceaue 30, and the thyrde 34, whiche ad­dyd togyther make 91 whiche is lesse by [...] then the sum 100, the which sholde be de­uided amonge them. Set then this fyrste false posycion 27 at the vpper ende of the crosse on the lefte syde, and the errour en­suynge of that at the foote of the same crosse on the same syde. And for bycause y^e this posicyon come to lesse then the verite therfore sette this letter L. for lesse, in the space betwene the posicion and y^e errour, as ye se in the example folowinge. Then coniecte agayn and suppose that the fyrst marchaunte had receaued 29 then by that rekennynge the seconde sholde receaue 3 [...] the thyrde 36, which all set togyther make [Page] 97, so that yet this posycion commeth not to the veryte 100, but lacketh 3 of it, wher­fore set this posyciō 29 at the vppar ende of the crosse at the ryghte syde, and this errour 3 at the foote of the crosse, and in the space betwene the posycion and the errour set this letter L, for lesse. Nowe for so much as bothe these posicyons be lesse then the veryte, worke as ye dyd before, accordynge to the rule, subduce the lesse errour 3 out of the greater errour 9, re­mayneth 6 for the dyuisor to be set in the space betwene the two errours. Thē mul­tiplye the fyrste posycion 27 by the second errour 2 cōmeth 81, then the seconde con­iecture 29 by the fyrst errour 9 cōmeth 261 then deducte 81 out of 261 remayneth 180 whiche diuyded by 6 the diuysor afore sayde, the quotient shalbe 30 whiche is the iust and very coniecture, the whiche ye sholde haue coniected. Thus ye haue had sufficient example of this fyrst rule of bothe more, and both lesse.

¶Here foloweth the rule of one more, and the other lesse.

WHen the one posycyon amounteth to more then the veryte, and the other lesse then the veryte, then adde the errours togyther and that addyd number shalbe the diuysor. Then multyplye the fyrste posycyon by the seconde errour, and the seconde posyon by the fyrste errour, and that that cōmeth of bothe these mul­typlycatours adde them togyther also, then diuyde this addyd nūber by the ad­dyd errours the dyuysor aforesayd, & the quocyent sheweth the true posycion.