| The residue | 1 | 2 | |||||
| The quocient | 2 | 0 | 0 | 4 | |||
| To be dyvydede | 8 | 8 | 6 | 3 | 7 | 0 | 4 |
| The dyvyser | 4 | 4 | 2 | 3 |
Another example. And me shalle not cesse fro suche settyng of figures forwarde, nether of settynge of þe quocient into the dyviser, neþer of subtraccioun of the dyvyser, tille the first of the dyvyser be with-draw fro þe first to be dividede. The whiche done, or ought,[17] oþer nought shalle remayne: and yf it be ought,[17] kepe it in the tables, And euer [vny] it to þe diviser. And yf þou wilt wete how many vnytees of þe divisioun Fol. 533. *wol growe to the nombre of the divisere, What the quotient shows. the nombre quocient wol shewe it: and whan suche divisioun is made, and þou [lust] prove yf thow have wele done or How to prove your division, no, Multiplie the quocient by the diviser, And the same figures wolle come ayene that thow haddest bifore and none other. And yf ought be residue, than with addicioun therof shalle come the same figures: And so multiplicacioun provithe divisioun, and dyvisioun multiplicacioun: or multiplication. as thus, yf multiplicacioun be made, divide it by the multipliant, and the nombre quocient wol shewe the nombre that was to be multipliede, etc.
Chapter VIII. Progression.
Definition of Progression. Progressioun is of nombre after [egalle] excesse fro oone or tweyne [take] [agregacioun]. of progressioun one is [naturelle] or contynuelle, þat oþer broken and discontynuelle. Natural Progression. Naturelle it is, whan me begynnethe with one, and kepethe ordure ouerlepyng one; as .1. 2. 3. 4. 5. 6., etc., so þat the nombre folowynge passithe the other be-fore in one. Broken Progression. Broken it is, whan me lepithe fro o nombre tille another, and kepithe not the contynuel ordire; as 1. 3. 5. 7. 9, etc. Ay me may begynne with .2., as þus; .2. 4. 6. 8., etc., and the nombre folowyng passethe the others by-fore by .2. And note wele, that naturelle progressioun ay begynnethe with one, and Intercise or broken progressioun, [omwhile] begynnythe with one, omwhile with twayne. Of progressioun naturell .2. rules ther be [yove], of the whiche the first is this; The 1st rule for Natural Progression. whan the progressioun naturelle endithe in even nombre, by the half therof multiplie þe next totalle ouerere nombre; Example of grace: .1. 2. 3. 4. Multiplie .5. by .2. and so .10. comethe of, that is the totalle nombre þerof. The second rule. The seconde rule is suche, whan the progressioun naturelle endithe in nombre ode. Take the more porcioun of the oddes, and multiplie therby the totalle nombre. Example of grace 1. 2. 3. 4. 5., multiplie .5. by .3, and thryes .5. shalle be resultant. so the nombre totalle is .15. The first rule of Broken Progression. Of progresioun [intercise], ther ben also .2.[18] rules; and þe first is þis: Whan the Intercise progression endithe in even nombre by half therof multiplie the next nombre to þat halfe as .2.[18] 4. 6. Multiplie .4. by .3. so þat is thryes .4., and .12. the nombre of alle the progressioun, wolle folow. The second rule. The seconde rule is this: whan the progressioun interscise endithe in ode, take þe more porcioun of alle þe nombre, Fol. 534. *and multiplie by hym-selfe; as .1. 3. 5. Multiplie .3. by hym-selfe, and þe some of alle wolle be .9., etc.
Chapter IX. Extraction of Roots.
The preamble of the extraction of roots. Here folowithe the extraccioun of rotis, and first in nombre quadrates. Wherfor me shalle se what is a [nombre quadrat], and what is the rote of a nombre quadrat, and what it is to draw out the rote of a nombre. And before other note this divisioun: Linear, superficial, and solid numbers. Of nombres one is [lyneal], anoþer [superficialle], anoþer quadrat, anoþer cubike or [hoole]. lyneal is that þat is considrede after the processe, havynge no respect to the direccioun of nombre in nombre, As a lyne hathe but one dymensioun that is to sey after the lengthe. Superficial numbers. Nombre superficial is þat comethe of ledynge of oo nombre into a-nother, wherfor it is callede superficial, for it hathe .2. nombres notyng or mesurynge hym, as a superficialle thynge hathe .2. dimensions, þat is to sey lengthe and brede. Square numbers. And for bycause a nombre may be hade in a-nother by .2. maners, þat is to sey other in hym-selfe, oþer in anoþer, Vnderstonde yf it be had in hym-self, It is a quadrat. ffor dyvisioun write by vnytes, hathe .4. sides even as a quadrangille. and yf the nombre be hade in a-noþer, the nombre is superficiel and not quadrat, as .2. hade in .3. makethe .6. that is þe first nombre superficielle; wherfor it is open þat alle nombre quadrat is superficiel, and not [conuertide]. The root of a square number. The rote of a nombre quadrat is þat nombre that is had of hym-self, as twies .2. makithe 4. and .4. is the first nombre quadrat, and 2. is his rote. 9. 8. 7. 6. 5. 4. 3. 2. 1. / The rote of the more quadrat .3. 1. 4. 2. 6. Notes of some examples of square roots here interpolated. The most nombre quadrat 9. 8. 7. 5. 9. 3. 4. 7. 6. / the remenent ouer the quadrat .6. 0. 8. 4. 5. / The first caas of nombre quadrat .5. 4. 7. 5. 6. The rote .2. 3. 4. The seconde caas .3. 8. 4. 5. The rote .6. 2. The thirde caas .2. 8. 1. 9. The rote .5. 3. The .4. caas .3. 2. 1. The rote .1. 7. / The 5. caas .9. 1. 2. 0. 4. / The rote 3. 0. 2. Solid numbers. The [solide nombre] or cubike is þat þat comytħe of double ledyng of nombre in nombre; Three dimensions of solids. And it is clepede a solide body that hathe þer-in .3 [dimensions] þat is to sey, lengthe, brede, and thiknesse. so þat nombre hathe .3. nombres to be brought forthe in hym. But nombre may be hade twies in nombre, for other it is hade in hym-selfe, oþer in a-noþer. Cubic numbers. If a nombre be hade twies in hym-self, oþer ones in his quadrat, þat is the same, þat a [cubike] Fol. 54. *is, And is the same that is solide. And yf a nombre twies be hade in a-noþer, the nombre is [clepede] solide and not cubike, as twies .3. and þat .2. makithe .12. All cubics are solid numbers. Wherfor it is [opyne] that alle cubike nombre is solide, and not [conuertide]. Cubike is þat nombre þat comythe of ledynge of hym-selfe twyes, or ones in his quadrat. And here-by it is open that o nombre is the [roote] of a quadrat and of a cubike. Natheles the same nombre is not quadrat and cubike. No number may be both linear and solid. Opyne it is also that alle nombres may be a rote to a quadrat and cubike, but not alle nombre quadrat or cubike. Therfor sithen þe ledynge of vnyte in hym-self ones or twies nought comethe but vnytes, Seithe Boice in Arsemetrike, Unity is not a number. that vnyte potencially is al nombre, and none in act. And vndirstonde wele also that betwix euery .2. quadrates ther is a meene proporcionalle, Examples of square roots. That is openede thus; [lede the rote of o quadrat into] the rote of the oþer quadrat, and þan wolle þe meene shew.
| Residuum | 0 | 4 | 0 | 0 | |||||||||||||
| Quadrande | 4 | 3 | 5 | 6 | 3 | 0 | 2 | 9 | 1 | 7 | 4 | 2 | 4 | 1 | 9 | 3 | 6 |
| Duplum | 1 | 2 | 1 | 0 | 2 | 6 | [8] | [19] | |||||||||
| Subduplum | 6 | 6 | 5 | 5 | 1 | 3 | 2 | 4 | 4 |
A note on mean proportionals. Also betwix the next .2. cubikis, me may fynde a double meene, that is to sey a more meene and a lesse. The more meene thus, as to brynge the rote of the lesse into a quadrat of the more. The lesse thus, If the rote of the more be brought Into the quadrat of the lesse.
Chapter X. Extraction of Square Root.
To[20] draw a rote of the nombre quadrat it is What-euer nombre be proposede to fynde his rote and to se yf it be quadrat. To find a square root. And yf it be not quadrat the rote of the most quadrat fynde out, vnder the nombre proposede. Therfor yf thow wilt the rote of any quadrat nombre draw out, write the nombre by his differences, and [compt] the nombre of the figures, and wete yf it be ode or even. And yf Begin with the last odd place. it be even, than most thow begynne worche vnder the last save one. And yf it be ode with the last; and forto sey it shortly, al-weyes fro the last ode me shalle begynne. Therfor vnder the last in an od place sette, Find the nearest square root of that number, subtract, me most fynde a digit, the whiche lade in hym-selfe it puttithe away that, þat is ouer his hede, oþer as neighe as me may: suche a digit founde and withdraw fro his ouerer, me most double that digit and sette the double vnder the next figure towarde the right honde, and his [vnder double] vnder hym. double it, That done, than me most fynde a-noþer digit vnder the next figure bifore the doublede, and set the double one to the right. the whiche Fol. 54 b. *brought in double settethe a-way alle that is ouer his hede as to [rewarde] of the doublede: Than brought into hym-self settithe all away in respect of hym-self, Find the second figure by division. Other do it as nye as it may be do: other me may with-draw the digit [21][last] founde, and lede hym in double or double hym, and after in hym-selfe; Multiply the double by the second figure, and add after it the square of the second figure, and subtract. Than Ioyne to-geder the produccione of them bothe, So that the first figure of the last product be addede before the first of the first productes, the seconde of the first, etc. and so forthe, [subtrahe] fro the totalle nombre in respect of þe digit.