The residue 5432
To be quadrede41209151399005432
The double 40 2 4 6 0 0
The vnder double2 0 31 2 3[3] [0] [0] 0

And if it hap þat no digit may be founde, Than sette a cifre vndre a cifre, and cesse not tille thow fynde a digit; and whan thow hast founde it to double it, neþer to sette the doublede forwarde nether the vnder doublede, Special cases. Till thow fynde vndre the first figure a digit, the whiche lade in alle double, settyng away alle that is ouer hym in respect of the doublede: Than lede hym into hym-selfe, and put a-way alle in regarde of hym, other as nyghe as thow maist. The residue. That done, other ought or nought wolle be the residue. If nought, than it shewithe that a nombre componede was the quadrat, and his rote a digit last founde with vndere-double other vndirdoubles, so that it be sette be-fore: And yf ought[22] remayne, that shewith that the nombre proposede was not quadrat,[23] but a digit [last found with the subduple or subduples is]

1234567 89
281216202428 3236
3182736455463 7281
43248648096112[24]128144
55075100125150175 200225
672108144180216252 288324
798147196245294343 393441
8128192256320384448 512576
9168243324405486567 648729[25]

The rote of the most quadrat conteynede vndre 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 founde with the vnder-double oþer vnder-doublis, and thow shalt fynde the same figures that thow haddest before; And so that nought be the Fol. 55. *residue. And yf thow have any residue, than with the addicioun þerof that is reseruede with-out in thy table, thow shalt fynde thi first figures as thow haddest them before, etc.

Chapter XI. Extraction of Cube Root.

Definition of a cubic number and a cube root. Heere folowithe the extraccioun of rotis in cubike nombres; 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 cubike it is, as it is before declarede, that comethe of ledyng of any nombre twies in hym-selfe, other ones in his quadrat. The rote of a nombre cubike is the nombre that is twies hade in hym-selfe, or ones in his quadrat. [Wher-thurghe] it is open, that euery nombre quadrat or cubike have the same rote, as it is seide before. And forto draw out the rote of a cubike, It is first to fynde þe nombre proposede yf it be a cubike; And yf it be not, than thow most make extraccioun of his rote of the most cubike vndre the nombre proposide his rote founde. Therfor proposede some 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 thousande place, thow most fynde a digit, the whiche lade in hym-self cubikly puttithe a-way that þat is ouer his hede as in respect of hym, other as nyghe as thow maist. treble it and place it under the next but one, and multiply by the digit. That done, thow most [trebille] the digit, and that triplat is to be put vnder the .3. next figure towarde the right honde, And the [vnder-trebille] vnder the trebille; Then find the second digit. Than me most fynde a digit vndre the next figure bifore the triplat, the whiche with his vnder-trebille had into a trebille, afterwarde other vnder[trebille][26] had in his produccioun, puttethe a-way alle that is ouer it in regarde of[27] [the triplat. Then lade in hymself puttithe away that þat is over his hede as in respect of hym, other as nyghe as thou maist:] Multiply the first triplate and the second digit, twice by this digit. That done, thow most trebille the digit ayene, and the triplat is to be sette vnder the next .3. figure as before, And the vnder-trebille vnder the trebille: and than most thow sette forwarde the first triplat with his vndre-trebille by .2. differences. And than most thow fynde a digit vnder the next figure before the triplat, the whiche withe his [vnder-triplat] had in his triplat afterwarde, Subtract. other vnder-treblis lad in product Fol. 55 b. *It sittethe a-way ałł that is ouer his hede in respect of the triplat than had in hym-self cubikly,[28] or as nyghe as ye may.

Residuum 5 4 1019
Cubicandus8365432 30076711667
Triplum 60 18 4
Subtriplum2 0 [3] 6 7 2 2

Continue this process till the first figure is reached. Nother me shalle not cesse of the fyndynge of that digit, neither of his triplacioun, neþer of the triplat-is [29][anterioracioun], that is to sey, settyng forwarde by .2. differences, Ne therof the vndre-triple to be put vndre the triple, Nether of the multiplicacioun þerof, Neither of the subtraccioun, tille it come to the first figure, vnder the whiche is a digitalle nombre to be founde, the whiche withe his vndre-treblis most be hade in tribles, After-warde without vnder-treblis to be hade into [produccioun], settyng away alle that is ouer the hede of the triplat nombre, After had into hymselfe cubikly, Examples. and sette alle-way that is ouer hym.

To be [cubicede]172832 768
The triple 32 9
The vnder triple 12 [3] 33

Also note wele that the produccion comynge of the ledyng of a digite founde[30] me may adde to, and also with-draw fro of the totalle nombre sette above that digit so founde.[31] The residue. That done ought or nought most be the residue. If it be nought, It is open that the nombre proposede was a cubike nombre, And his rote a digit founde last with the vnder-triples: If the rote therof [wex] bade in hym-selfe, and afterwarde product they shalle make the first figures. And yf ought be in residue, kepe that without in the table; and it is opene that the nombre was not a cubike. but a digit last founde with the vndirtriplis is rote of the most cubike vndre the nombre proposede conteynede, the whiche rote yf it be hade in hym-selfe, Special cases. And afterwarde in a product of that shalle growe the most cubike vndre the nombre proposede conteynede, And yf that be addede to a cubike the residue reseruede in the table, wolle make the same figures that ye hade first. Special case. Fol. 56. *And yf no digit after the anterioracioun[32] may not be founde, than put there a cifre vndre a cifre vndir the thirde figure, And put forwarde þe figures. Note also wele that yf in the nombre proposede ther ben no place of thowsandes, me most begynne vnder the first figure in the extraccioun of the rote. some vsen forto [distingue] the nombre by threes, and ay begynne forto wirche vndre the first of the last [ternary] other uncomplete nombre, the whiche maner of operacioun accordethe with that before.