Chapter X

. Extraction of Square Root.]

[Sidenote: To find a square root. Begin with the last odd place. Find the nearest square root of that number, subtract, double it, and set the double one to the right. Find the second figure by division. Multiply the double by the second figure, and add after it the square of the second figure, and subtract.]

[{20}]To draw a rote of the nombre quadrat it is What-eu{er} nombre be p{ro}posed{e} to fynde his rote and to se yf it be quadrat. And yf it be not quadrat the rote of the most quadrat fynde out, vnder the nombre p{ro}posed{e}. 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 od{e} or even. And yf it be even, than most thow begynne worche vnder the last save one. And yf it be od{e} w{i}t{h} the last; and forto sey it shortly, al-weyes fro the last od{e} me shall{e} begynne. Therfor vnder the last in an od place sette, me most fynd{e} a digit, the which{e} lad{e} in hym-self{e} it puttith{e} away that, þat is ou{er} his hede, oþ{er} as neigh{e} as me may: suche a digit found{e} and w{i}t{h}draw fro his ou{er}er, me most double that digit and sette the double vnder the next figure toward{e} the right hond{e}, and his vnder double vnder hym. That done, than me most fy{n}d{e} a-noþ{er} digit vnder the next figure bifore the doubled{e}, the which{e} [*Fol. 54b] brought in double setteth{e} a-way all{e} that is ou{er} his hede as to reward{e} of the doubled{e}: Than brought into hym-self settith{e} all away in respect of hym-self, Other do it as nye as it may be do: other me may w{i}t{h}-draw the digit [{21}][last] found{e}, and lede hym in double or double hym, and after in hym-self{e}; Than Ioyne to-geder the p{ro}duccion{e} of them bothe, So that the first figure of the last p{ro}duct be added{e} before the first of the first p{ro}duct{es}, the second{e} of the first, {et}c. and so forth{e}, subtrahe fro the totall{e} nombre in respect of þe digit.

[Sidenote: Examples.]

+------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+ | The residue | | | | | || | | | | || | | |5| 4 |3|2| +------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+ | To be quadred{e} |4|1|2|0|9||1|5|1|3|9|| 9 |0| 0 |5| 4 |3|2| +------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+ | The double | |4|0| | || |2| |4| || |6| |0| | |0| +------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+ | The vnder double |2| |0| |3||1| |2| |3||[3]| |[0]| |[0]| |0| +------------------+-+-+-+-+-++-+-+-+-+-++---+-+---+-+---+-+-+

[Sidenote: Special cases. The residue.]

And if it hap þ{a}t no digit may be found{e}, Than sette a cifre vndre a cifre, and cesse not till{e} thow fynde a digit; and whan thow hast founde it to double it, neþ{er} to sette the doubled{e} forward{e} nether the vnder doubled{e}, Till thow fynde vndre the first figure a digit, the which{e} lad{e} in all{e} double, settyng away all{e} that is ou{er} hym in respect of the doubled{e}: Than lede hym into hym-self{e}, and put a-way all{e} in regard{e} of hym, other as nygh{e} as thow maist. That done, other ought or nought wolle be the residue. If nought, than it shewith{e} that a nombre componed{e} was the quadrat, and his rote a digit last found{e} w{i}t{h} vnder{e}-double other vndirdoubles, so that it be sette be-fore: And yf ought[{22}] remayn{e}, that shew{i}t{h} that the nombre p{ro}posed{e} was not quadrat,[{23}] [[wher-vpon{e} se the table in the next side of the next leef{e}.]] but a digit [last found with the subduple or subduples is]

[Sidenote: This table is constructed for use in cube root sums, giving the value of ab.^2]

+---+-----+-----+-----+-----+-----+---------+-----+---------+ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | +---+-----+-----+-----+-----+-----+---------+-----+---------+ | 2 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | +---+-----+-----+-----+-----+-----+---------+-----+---------+ | 3 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | +---+-----+-----+-----+-----+-----+---------+-----+---------+ | 4 | 32 | 48 | 64 | 80 | 96 |112[{24}]| 128 | 144 | +---+-----+-----+-----+-----+-----+---------+-----+---------+ | 5 | 50 | 75 | 100 | 125 | 150 | 175 | 200 | 225 | +---+-----+-----+-----+-----+-----+---------+-----+---------+ | 6 | 72 | 108 | 144 | 180 | 216 | 252 | 288 | 324 | +---+-----+-----+-----+-----+-----+---------+-----+---------+ | 7 | 98 | 147 | 196 | 245 | 294 | 343 | 393 | 441 | +---+-----+-----+-----+-----+-----+---------+-----+---------+ | 8 | 128 | 192 | 256 | 320 | 384 | 448 | 512 | 576 | +---+-----+-----+-----+-----+-----+---------+-----+---------+ | 9 | 168 | 243 | 324 | 405 | 486 | 567 | 648 |729[{25}]| +---+-----+-----+-----+-----+-----+---------+-----+---------+

[Sidenote: How to prove the square root without or with a remainder.]

The rote of the most quadrat conteyned{e} vndre the nombre p{ro}posed{e}. Therfor yf thow wilt p{ro}ve yf thow have wele do or no, Multiplie the digit last found{e} w{i}t{h} the vnder-double oþ{er} 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 w{i}t{h} the addicio{u}n þ{er}of that is res{er}ued{e} w{i}t{h}-out in thy table, thow shalt fynd{e} thi first figures as thow haddest them before, {et}c.

[Headnote: