And if it hap þat no digit may be founde, Than sette a cifre vndre a cifre, and cesse not tillethow fynde a digit; and whan thow hast founde it to double it, neþerto sette the doubledeforwardenether the vnder doublede,Special cases.Till thow fynde vndre the first figure a digit, the whicheladein alledouble, settyng away allethat is ouerhym in respect of the doublede: Than lede hym into hym-selfe, and put a-way allein regardeof hym, other as nygheas thow maist.The residue.That done, other ought or nought wolle be the residue. If nought, than it shewithethat a nombre componedewas the quadrat, and his rote a digit last foundewithvndere-double other vndirdoubles, so that it be sette be-fore: And yf ought22remayne, that shewiththat the nombre proposedewas not quadrat,23but a digit [last found with the subduple or subduplesis]This table is constructed for use in cube root sums, giving the value ofab.21234567892812162024283236318273645546372814324864809611224128144550751001251501752002256721081441802162522883247981471962452943433934418128192256320384448512576916824332440548656764872925The rote of the most quadrat conteynedevndre the nombre proposede.How to prove the square root without or with a remainder.Therfor yf thow wilt prove yf thow have wele do or no, Multiplie the digit last foundewiththe vnder-double oþervnder-doublis, and thow shalt fynde the same figures that thow haddest before; And so that nought be theFol. 55.*residue. And yf thow have any residue, than withthe addicioun þerof that is reseruedewith-out in thy table, thow shalt fyndethi first figures as thow haddest them before,etc.Chapter XI. Extraction of Cube Root.Definition of a cubic number and a cube root.Heere folowithethe extraccioun of rotis in cubikenombres; wher-for me most se what is a nombre cubike, and what is his roote, And what is the extraccioun of a rote. A nombre cubikeit is, as it is before declarede, that cometheof ledyng of any nombre twies in hym-selfe, other ones in his quadrat. The rote of a nombre cubikeis the nombre that is twies hadein hym-selfe, or ones in his quadrat.Wher-thurgheit is open, that euery nombre quadrat or cubikehave the same rote, as it is seidebefore. And forto draw out the rote of a cubike, It is first to fyndeþe nombreproposedeyf it be a cubike; And yf it be not, than thow most make extraccioun of his rote of the most cubikevndre the nombre proposidehis rote founde. Therfor proposedesome nombre, whos cubical rote þou woldest draw out;Mark off the places in threes.First thow most compt the figures by fourthes, that is to sey in the place of thousandes;Find the first digit;And vnder the last thousandeplace, thow most fynde a digit, the whicheladein hym-self cubikly puttithea-way that þat is ouerhis hede as in respect of hym, other as nygheas thow maist.treble it and place it under the next but one, and multiply by the digit.That done, thow mosttrebillethe digit, and that triplat is to be put vnder the .3. next figure towardethe right honde, And thevnder-trebillevnder the trebille;Then find the second digit.Than me most fyndea digit vndre the next figure bifore the triplat, the whichewithhis vnder-trebillehad into a trebille, afterwarde other vnder[trebille]26had in his produccioun, puttethea-way allethat is ouerit in regardeof27[the triplat. Then lade in hymself puttithe away that þat is over his hede as in respect of hym, other as nyghe as thou maist:]Multiply the first triplate and the second digit, twice by this digit.That done, thow most trebillethe digit ayene, and the triplat is to be sette vnder the next .3. figure as before, And the vnder-trebillevnder the trebille: and than most thow sette forwardethe first triplat withhis vndre-trebilleby .2. differences. And than most thow fynde a digit vnder the next figure before the triplat, the whichewithehisvnder-triplathad in his triplat afterwarde,Subtract.other vnder-treblis lad in productFol. 55b.*It sittethea-way ałł that is ouerhis hede in respect of the triplat than had in hym-self cubikly,28or as nygheas ye may.Examples.Residuum541019Cubicandus836543230076711667Triplum60184Subtriplum20[3]6722Continue this process till the first figure is reached.Nother me shallenot cesse of the fyndyngeof that digit, neither of his triplacioun, neþerof the triplat-is29anterioracioun, that is to sey, settyng forwardeby .2. differences, Ne therof the vndre-triple to be put vndre the triple, Nether of the multiplicacioun þerof, Neither of the subtraccioun, tilleit come to the first figure, vnder the whicheis a digitallenombre to be founde, the whichewithehis vndre-treblis most be hadein tribles, After-wardewithout vnder-treblis to be hadeintoproduccioun, settyng away allethat is ouerthe hedeof the triplat nombre, After had into hymselfecubikly,Examples.and sette alle-way that is ouerhym.To becubicede172832768The triple329The vnder triple12[3]33Also note wele that the produccion comyngeof the ledyng of a digite founde30me may adde to, and also with-draw fro of the totallenombre sette above that digit so founde.31The residue.That done ought or nought most be the residue. If it be nought, It is open that the nombre proposedewas a cubikenombre, And his rote a digit founde last withthe vnder-triples: If the rote therofwexbadein hym-selfe, and afterwardeproduct they shallemake the first figures. And yf ought be in residue, kepe that without in the table; and it is openethat the nombre was not a cubike. but a digit last founde withthe vndirtriplis is rote of the most cubikevndre the nombre proposedeconteynede, the whicherote yf it be hadein hym-selfe,Special cases.And afterwardein a product of that shallegrowe the most cubikevndre the nombre proposedeconteynede, And yf that be addedeto a cubikethe residue reseruedein the table, wollemake the same figures that ye hadefirst.Special case.Fol. 56.*Andyf no digit after the anterioracioun32may not be founde, than put therea cifre vndre a cifre vndir the thirdefigure, And put forwardeþe figures. Note also wele that yf in the nombre proposedether ben no place of thowsandes, me most begynne vnder the first figure in the extraccioun of the rote. some vsen fortodistinguethe nombre by threes, and ay begynne forto wirchevndre the first of the lastternaryother uncomplete nombre, the whichemaner of operacioun accordethewiththat before.Examples.The residue011The cubicandus80000008242419The triple33006The vndertriple[2]00262And this at this tyme suffisethein extraccioun of nombres quadrat or cubikesetc.Table of Numbers, &c.A table of numbers; probably from the Abacus.12345one.x.an. hundrede/a thowsande/x. thowsande/67An hundredethowsande/A thowsandetymes a thowsande/x. thousandetymes a thousande/ An hundredethousandetymes a thousandeA thousandethousandetymes a thousande/ this is the x placeetc.[Ende.]1.MS. Materialle.2.MS. Formalle.3.‘the’ in MS.4.‘be’ in MS.5.‘and’ in MS.6.‘is’ in MS.7.6 in MS.8.0 in MS.9.2 in MS.10.sic.11.‘And’ inserted in MS.12.‘4 the’ inserted in MS.13.‘to’ in MS.14.‘that’ repeated in MS.15.‘1’ in MS.16.Blank in MS.17.‘nought’ in MS.18.3 written for 2 in MS.19.7 in MS.20.runs on in MS.21.‘so’ in MS.22.‘nought’ in MS.23.MS. adds here: ‘wher-vponese the table in the next side of the next leefe.’24.110 in MS.25.0 in MS.26.double in MS.27.‘it hym-selfe’ in MS.28.MS. adds here: ‘it settethea-way allehis respect.’29.‘aucterioracioun’ in MS.30.MS. adds here: ’withan vndre-triple / other of an vndre-triple in a triple or triplat is And after-wardewithout vndre-triple other vndre-triplis in the product and ayene that product that cometheof the ledyngeof a digit foundein hym-selfecubicalle’ /31.MS. adds here: ‘as ther had be a divisioun made as it is openedebefore.’32.MS. anteriocacioun.33.4 in MS.
And if it hap þat no digit may be founde, Than sette a cifre vndre a cifre, and cesse not tillethow fynde a digit; and whan thow hast founde it to double it, neþerto sette the doubledeforwardenether the vnder doublede,Special cases.Till thow fynde vndre the first figure a digit, the whicheladein alledouble, settyng away allethat is ouerhym in respect of the doublede: Than lede hym into hym-selfe, and put a-way allein regardeof hym, other as nygheas thow maist.The residue.That done, other ought or nought wolle be the residue. If nought, than it shewithethat a nombre componedewas the quadrat, and his rote a digit last foundewithvndere-double other vndirdoubles, so that it be sette be-fore: And yf ought22remayne, that shewiththat the nombre proposedewas not quadrat,23but a digit [last found with the subduple or subduplesis]This table is constructed for use in cube root sums, giving the value ofab.21234567892812162024283236318273645546372814324864809611224128144550751001251501752002256721081441802162522883247981471962452943433934418128192256320384448512576916824332440548656764872925The rote of the most quadrat conteynedevndre the nombre proposede.How to prove the square root without or with a remainder.Therfor yf thow wilt prove yf thow have wele do or no, Multiplie the digit last foundewiththe vnder-double oþervnder-doublis, and thow shalt fynde the same figures that thow haddest before; And so that nought be theFol. 55.*residue. And yf thow have any residue, than withthe addicioun þerof that is reseruedewith-out in thy table, thow shalt fyndethi first figures as thow haddest them before,etc.Chapter XI. Extraction of Cube Root.Definition of a cubic number and a cube root.Heere folowithethe extraccioun of rotis in cubikenombres; wher-for me most se what is a nombre cubike, and what is his roote, And what is the extraccioun of a rote. A nombre cubikeit is, as it is before declarede, that cometheof ledyng of any nombre twies in hym-selfe, other ones in his quadrat. The rote of a nombre cubikeis the nombre that is twies hadein hym-selfe, or ones in his quadrat.Wher-thurgheit is open, that euery nombre quadrat or cubikehave the same rote, as it is seidebefore. And forto draw out the rote of a cubike, It is first to fyndeþe nombreproposedeyf it be a cubike; And yf it be not, than thow most make extraccioun of his rote of the most cubikevndre the nombre proposidehis rote founde. Therfor proposedesome nombre, whos cubical rote þou woldest draw out;Mark off the places in threes.First thow most compt the figures by fourthes, that is to sey in the place of thousandes;Find the first digit;And vnder the last thousandeplace, thow most fynde a digit, the whicheladein hym-self cubikly puttithea-way that þat is ouerhis hede as in respect of hym, other as nygheas thow maist.treble it and place it under the next but one, and multiply by the digit.That done, thow mosttrebillethe digit, and that triplat is to be put vnder the .3. next figure towardethe right honde, And thevnder-trebillevnder the trebille;Then find the second digit.Than me most fyndea digit vndre the next figure bifore the triplat, the whichewithhis vnder-trebillehad into a trebille, afterwarde other vnder[trebille]26had in his produccioun, puttethea-way allethat is ouerit in regardeof27[the triplat. Then lade in hymself puttithe away that þat is over his hede as in respect of hym, other as nyghe as thou maist:]Multiply the first triplate and the second digit, twice by this digit.That done, thow most trebillethe digit ayene, and the triplat is to be sette vnder the next .3. figure as before, And the vnder-trebillevnder the trebille: and than most thow sette forwardethe first triplat withhis vndre-trebilleby .2. differences. And than most thow fynde a digit vnder the next figure before the triplat, the whichewithehisvnder-triplathad in his triplat afterwarde,Subtract.other vnder-treblis lad in productFol. 55b.*It sittethea-way ałł that is ouerhis hede in respect of the triplat than had in hym-self cubikly,28or as nygheas ye may.Examples.Residuum541019Cubicandus836543230076711667Triplum60184Subtriplum20[3]6722Continue this process till the first figure is reached.Nother me shallenot cesse of the fyndyngeof that digit, neither of his triplacioun, neþerof the triplat-is29anterioracioun, that is to sey, settyng forwardeby .2. differences, Ne therof the vndre-triple to be put vndre the triple, Nether of the multiplicacioun þerof, Neither of the subtraccioun, tilleit come to the first figure, vnder the whicheis a digitallenombre to be founde, the whichewithehis vndre-treblis most be hadein tribles, After-wardewithout vnder-treblis to be hadeintoproduccioun, settyng away allethat is ouerthe hedeof the triplat nombre, After had into hymselfecubikly,Examples.and sette alle-way that is ouerhym.To becubicede172832768The triple329The vnder triple12[3]33Also note wele that the produccion comyngeof the ledyng of a digite founde30me may adde to, and also with-draw fro of the totallenombre sette above that digit so founde.31The residue.That done ought or nought most be the residue. If it be nought, It is open that the nombre proposedewas a cubikenombre, And his rote a digit founde last withthe vnder-triples: If the rote therofwexbadein hym-selfe, and afterwardeproduct they shallemake the first figures. And yf ought be in residue, kepe that without in the table; and it is openethat the nombre was not a cubike. but a digit last founde withthe vndirtriplis is rote of the most cubikevndre the nombre proposedeconteynede, the whicherote yf it be hadein hym-selfe,Special cases.And afterwardein a product of that shallegrowe the most cubikevndre the nombre proposedeconteynede, And yf that be addedeto a cubikethe residue reseruedein the table, wollemake the same figures that ye hadefirst.Special case.Fol. 56.*Andyf no digit after the anterioracioun32may not be founde, than put therea cifre vndre a cifre vndir the thirdefigure, And put forwardeþe figures. Note also wele that yf in the nombre proposedether ben no place of thowsandes, me most begynne vnder the first figure in the extraccioun of the rote. some vsen fortodistinguethe nombre by threes, and ay begynne forto wirchevndre the first of the lastternaryother uncomplete nombre, the whichemaner of operacioun accordethewiththat before.Examples.The residue011The cubicandus80000008242419The triple33006The vndertriple[2]00262And this at this tyme suffisethein extraccioun of nombres quadrat or cubikesetc.Table of Numbers, &c.A table of numbers; probably from the Abacus.12345one.x.an. hundrede/a thowsande/x. thowsande/67An hundredethowsande/A thowsandetymes a thowsande/x. thousandetymes a thousande/ An hundredethousandetymes a thousandeA thousandethousandetymes a thousande/ this is the x placeetc.[Ende.]1.MS. Materialle.2.MS. Formalle.3.‘the’ in MS.4.‘be’ in MS.5.‘and’ in MS.6.‘is’ in MS.7.6 in MS.8.0 in MS.9.2 in MS.10.sic.11.‘And’ inserted in MS.12.‘4 the’ inserted in MS.13.‘to’ in MS.14.‘that’ repeated in MS.15.‘1’ in MS.16.Blank in MS.17.‘nought’ in MS.18.3 written for 2 in MS.19.7 in MS.20.runs on in MS.21.‘so’ in MS.22.‘nought’ in MS.23.MS. adds here: ‘wher-vponese the table in the next side of the next leefe.’24.110 in MS.25.0 in MS.26.double in MS.27.‘it hym-selfe’ in MS.28.MS. adds here: ‘it settethea-way allehis respect.’29.‘aucterioracioun’ in MS.30.MS. adds here: ’withan vndre-triple / other of an vndre-triple in a triple or triplat is And after-wardewithout vndre-triple other vndre-triplis in the product and ayene that product that cometheof the ledyngeof a digit foundein hym-selfecubicalle’ /31.MS. adds here: ‘as ther had be a divisioun made as it is openedebefore.’32.MS. anteriocacioun.33.4 in MS.
And if it hap þat no digit may be founde, Than sette a cifre vndre a cifre, and cesse not tillethow fynde a digit; and whan thow hast founde it to double it, neþerto sette the doubledeforwardenether the vnder doublede,Special cases.Till thow fynde vndre the first figure a digit, the whicheladein alledouble, settyng away allethat is ouerhym in respect of the doublede: Than lede hym into hym-selfe, and put a-way allein regardeof hym, other as nygheas thow maist.The residue.That done, other ought or nought wolle be the residue. If nought, than it shewithethat a nombre componedewas the quadrat, and his rote a digit last foundewithvndere-double other vndirdoubles, so that it be sette be-fore: And yf ought22remayne, that shewiththat the nombre proposedewas not quadrat,23but a digit [last found with the subduple or subduplesis]
The rote of the most quadrat conteynedevndre the nombre proposede.How to prove the square root without or with a remainder.Therfor yf thow wilt prove yf thow have wele do or no, Multiplie the digit last foundewiththe vnder-double oþervnder-doublis, and thow shalt fynde the same figures that thow haddest before; And so that nought be theFol. 55.*residue. And yf thow have any residue, than withthe addicioun þerof that is reseruedewith-out in thy table, thow shalt fyndethi first figures as thow haddest them before,etc.
Chapter XI. Extraction of Cube Root.
Definition of a cubic number and a cube root.Heere folowithethe extraccioun of rotis in cubikenombres; wher-for me most se what is a nombre cubike, and what is his roote, And what is the extraccioun of a rote. A nombre cubikeit is, as it is before declarede, that cometheof ledyng of any nombre twies in hym-selfe, other ones in his quadrat. The rote of a nombre cubikeis the nombre that is twies hadein hym-selfe, or ones in his quadrat.Wher-thurgheit is open, that euery nombre quadrat or cubikehave the same rote, as it is seidebefore. And forto draw out the rote of a cubike, It is first to fyndeþe nombreproposedeyf it be a cubike; And yf it be not, than thow most make extraccioun of his rote of the most cubikevndre the nombre proposidehis rote founde. Therfor proposedesome nombre, whos cubical rote þou woldest draw out;Mark off the places in threes.First thow most compt the figures by fourthes, that is to sey in the place of thousandes;Find the first digit;And vnder the last thousandeplace, thow most fynde a digit, the whicheladein hym-self cubikly puttithea-way that þat is ouerhis hede as in respect of hym, other as nygheas thow maist.treble it and place it under the next but one, and multiply by the digit.That done, thow mosttrebillethe digit, and that triplat is to be put vnder the .3. next figure towardethe right honde, And thevnder-trebillevnder the trebille;Then find the second digit.Than me most fyndea digit vndre the next figure bifore the triplat, the whichewithhis vnder-trebillehad into a trebille, afterwarde other vnder[trebille]26had in his produccioun, puttethea-way allethat is ouerit in regardeof27[the triplat. Then lade in hymself puttithe away that þat is over his hede as in respect of hym, other as nyghe as thou maist:]Multiply the first triplate and the second digit, twice by this digit.That done, thow most trebillethe digit ayene, and the triplat is to be sette vnder the next .3. figure as before, And the vnder-trebillevnder the trebille: and than most thow sette forwardethe first triplat withhis vndre-trebilleby .2. differences. And than most thow fynde a digit vnder the next figure before the triplat, the whichewithehisvnder-triplathad in his triplat afterwarde,Subtract.other vnder-treblis lad in productFol. 55b.*It sittethea-way ałł that is ouerhis hede in respect of the triplat than had in hym-self cubikly,28or as nygheas ye may.
Continue this process till the first figure is reached.Nother me shallenot cesse of the fyndyngeof that digit, neither of his triplacioun, neþerof the triplat-is29anterioracioun, that is to sey, settyng forwardeby .2. differences, Ne therof the vndre-triple to be put vndre the triple, Nether of the multiplicacioun þerof, Neither of the subtraccioun, tilleit come to the first figure, vnder the whicheis a digitallenombre to be founde, the whichewithehis vndre-treblis most be hadein tribles, After-wardewithout vnder-treblis to be hadeintoproduccioun, settyng away allethat is ouerthe hedeof the triplat nombre, After had into hymselfecubikly,Examples.and sette alle-way that is ouerhym.
Also note wele that the produccion comyngeof the ledyng of a digite founde30me may adde to, and also with-draw fro of the totallenombre sette above that digit so founde.31The residue.That done ought or nought most be the residue. If it be nought, It is open that the nombre proposedewas a cubikenombre, And his rote a digit founde last withthe vnder-triples: If the rote therofwexbadein hym-selfe, and afterwardeproduct they shallemake the first figures. And yf ought be in residue, kepe that without in the table; and it is openethat the nombre was not a cubike. but a digit last founde withthe vndirtriplis is rote of the most cubikevndre the nombre proposedeconteynede, the whicherote yf it be hadein hym-selfe,Special cases.And afterwardein a product of that shallegrowe the most cubikevndre the nombre proposedeconteynede, And yf that be addedeto a cubikethe residue reseruedein the table, wollemake the same figures that ye hadefirst.Special case.Fol. 56.*Andyf no digit after the anterioracioun32may not be founde, than put therea cifre vndre a cifre vndir the thirdefigure, And put forwardeþe figures. Note also wele that yf in the nombre proposedether ben no place of thowsandes, me most begynne vnder the first figure in the extraccioun of the rote. some vsen fortodistinguethe nombre by threes, and ay begynne forto wirchevndre the first of the lastternaryother uncomplete nombre, the whichemaner of operacioun accordethewiththat before.
And this at this tyme suffisethein extraccioun of nombres quadrat or cubikesetc.
Table of Numbers, &c.
x. thousandetymes a thousande/ An hundredethousandetymes a thousandeA thousandethousandetymes a thousande/ this is the x placeetc.
[Ende.]
1.MS. Materialle.2.MS. Formalle.3.‘the’ in MS.4.‘be’ in MS.5.‘and’ in MS.6.‘is’ in MS.7.6 in MS.8.0 in MS.9.2 in MS.10.sic.11.‘And’ inserted in MS.12.‘4 the’ inserted in MS.13.‘to’ in MS.14.‘that’ repeated in MS.15.‘1’ in MS.16.Blank in MS.17.‘nought’ in MS.18.3 written for 2 in MS.19.7 in MS.20.runs on in MS.21.‘so’ in MS.22.‘nought’ in MS.23.MS. adds here: ‘wher-vponese the table in the next side of the next leefe.’24.110 in MS.25.0 in MS.26.double in MS.27.‘it hym-selfe’ in MS.28.MS. adds here: ‘it settethea-way allehis respect.’29.‘aucterioracioun’ in MS.30.MS. adds here: ’withan vndre-triple / other of an vndre-triple in a triple or triplat is And after-wardewithout vndre-triple other vndre-triplis in the product and ayene that product that cometheof the ledyngeof a digit foundein hym-selfecubicalle’ /31.MS. adds here: ‘as ther had be a divisioun made as it is openedebefore.’32.MS. anteriocacioun.33.4 in MS.
1.MS. Materialle.
2.MS. Formalle.
3.‘the’ in MS.
4.‘be’ in MS.
5.‘and’ in MS.
6.‘is’ in MS.
7.6 in MS.
8.0 in MS.
9.2 in MS.
10.sic.
11.‘And’ inserted in MS.
12.‘4 the’ inserted in MS.
13.‘to’ in MS.
14.‘that’ repeated in MS.
15.‘1’ in MS.
16.Blank in MS.
17.‘nought’ in MS.
18.3 written for 2 in MS.
19.7 in MS.
20.runs on in MS.
21.‘so’ in MS.
22.‘nought’ in MS.
23.MS. adds here: ‘wher-vponese the table in the next side of the next leefe.’
24.110 in MS.
25.0 in MS.
26.double in MS.
27.‘it hym-selfe’ in MS.
28.MS. adds here: ‘it settethea-way allehis respect.’
29.‘aucterioracioun’ in MS.
30.MS. adds here: ’withan vndre-triple / other of an vndre-triple in a triple or triplat is And after-wardewithout vndre-triple other vndre-triplis in the product and ayene that product that cometheof the ledyngeof a digit foundein hym-selfecubicalle’ /
31.MS. adds here: ‘as ther had be a divisioun made as it is openedebefore.’
32.MS. anteriocacioun.
33.4 in MS.