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 to sey, length{e}, brede, and thiknesse. so þ{a}t nombre hath{e} .3. nombres to be brought forth{e} in hym. But nombre may be had{e} twies in nombre, for other it is had{e} in hym-self{e}, oþ{er} in a-noþ{er}. If a nombre be had{e} twies in hym-self, oþ{er} ones in his quadrat, þ{a}t is the same, þ{a}t a cubik{e} [*Fol. 54.] is, And is the same that is solide. And yf a nombre twies be had{e} in a-noþ{er}, the nombre is cleped{e} solide and not cubik{e}, as twies .3. and þ{a}t .2. makith{e} .12. Wherfor it is opyn{e} that all{e} cubik{e} nombre is solid{e}, and not {con}u{er}tid{e}. Cubik{e} is þ{a}t nombre þat comyth{e} of ledyng{e} of hym-self{e} twyes, or ones in his quadrat. And here-by it is open that o nombre is the roote of a quadrat and of a cubik{e}. Natheles the same nombre is not q{ua}drat and cubik{e}. Opyn{e} it is also that all{e} nombres may be a rote to a q{ua}drat and cubik{e}, but not all{e} nombre quadrat or cubik{e}. Therfor sithen þe ledyng{e} of vnyte in hym-self ones or twies nought cometh{e} but vnytes, Seith{e} Boice in Arsemetrik{e}, that vnyte potencially is al nombre, and none in act. And vndirstond{e} wele also that betwix euery .2. quadrat{es} ther is a meene p{ro}porcionall{e}, That is opened{e} thus; lede the rote of o quadrat into the rote of the oþ{er} quadrat, and þan wolle þe meene shew.

[Sidenote: Examples of square roots.]

+-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+ | Residuu{m} | | |0| || | | |4|| | |0| | || | | 0 | | +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+ | Quadrand{e} |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}]| | +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+ | Subduplu{m} | |6| |6|| |5| |5||1| |3| |2|| | 4 | |4| +-------------+-+-+-+-++-+-+-+-++-+-+-+-+-++-+---+------+-+

[Sidenote: 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 bryng{e} 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.

[Headnote: