half therof multiplie the next nombre to þat half{e} as .2.[{18}] 4. 6. Multiplie .4. by .3. so þat is thryes .4., and .12. the nombre of all{e} the p{ro}gressio{u}n, woll{e} folow. The second{e} rule is this: whan the p{ro}gressio{u}n int{er}scise endith{e} in od{e}, take þe more porcio{u}n of all{e} þe nombre, [*Fol. 53^4.] and multiplie by hym-self{e}; as .1. 3. 5. Multiplie .3. by hym-self{e}, and þe some of all{e} wolle be .9., {et}c. [Headnote: Chapter IX. Extraction of Roots.] [Sidenote: The preamble of the extraction of roots. Linear, superficial, and solid numbers. Superficial numbers. Square numbers. The root of a square number. Notes of some examples of square roots here interpolated. Solid numbers. Three dimensions of solids. Cubic numbers. All cubics are solid numbers. No number may be both linear and solid. Unity is not a number.] Here folowith{e} the extraccio{u}n of rotis, and first in nombre q{ua}drat{es}. Wherfor me shall{e} se what is a nombre quadrat, and what is the rote of a nombre quadrat, and what it is to draw out the rote of a nombre. And before other note this divisio{u}n: Of nombres one is lyneal, anoþ{er} sup{er}ficiall{e}, anoþ{er} quadrat, anoþ{er} cubik{e} or hoole. lyneal is that þat is considred{e} after the p{ro}cesse, havyng{e} no respect to the direccio{u}n of nombre in nombre, As a lyne hath{e} but one dymensio{u}n that is to sey after the length{e}. Nombre sup{er}ficial is þ{a}t cometh{e} of ledyng{e} of oo nombre into a-nother, wherfor it is called{e} sup{er}ficial, for it hath{e} .2. nombres notyng or mesuryng{e} hym, as a sup{er}ficiall{e} thyng{e} hath{e} .2. dimensions, þ{a}t is to sey length{e} and brede. And for bycause a nombre may be had{e} in a-nother by .2. man{er}s, þ{a}t is to sey other in hym-self{e}, oþ{er} in anoþ{er}, Vnderstond{e} yf it be had in hym-self, It is a quadrat. ffor dyvisio{u}n write by vnytes, hath{e} .4. sides even as a quadrangill{e}. and yf the nombre be had{e} in a-noþ{er}, the nombre is sup{er}ficiel and not quadrat, as .2. had{e} in .3. maketh{e} .6. that is þe first nombre sup{er}ficiell{e}; wherfor it is open þat all{e} nombre quadrat is sup{er}ficiel, and not co{n}u{er}tid{e}. The rote of a nombre quadrat is þat nombre that is had of hym-self, as twies .2. makith{e} 4. and .4. is the first nombre quadrat, and 2. is his rote. 9. 8. 7. 6. 5. 4. 3. 2. 1. / The rote of the more quadrat .3. 1. 4. 2. 6. The most nombre quadrat 9. 8. 7. 5. 9. 3. 4. 7. 6. / the remenent ou{er} the quadrat .6. 0. 8. 4. 5. / The first caas of nombre quadrat .5. 4. 7. 5. 6. The rote .2. 3. 4. The second{e} caas .3. 8. 4. 5. The rote .6. 2. The third{e} caas .2. 8. 1. 9. The rote .5. 3. The .4. caas .3. 2. 1. The rote .1. 7. / The 5. caas .9. 1. 2. 0. 4. / The rote 3. 0. 2. The solid{e} nombre or cubik{e} is þat þ{a}t comytħe of double ledyng of nombre in nombre; And it is cleped{e} a solid{e} body that hath{e} þ{er}-in .3 [dimensions] þat is