Page:Encyclopædia Britannica, first edition - Volume I, A-B.pdf/131

 I, G E n B R A. 99 Dem. an--nz+Xtn'T1mbi-)ran-r*r»li‘llz4zat,—Amb*i . . . . artheifibinomial g to/7/ fund, be thatexprejfed of the propofd quantity in its leaf terms.divided Thus,by 0 —i>0 J _ 3 V' 5 + g /2 _ = ^5+^2; Xa«=±=i5{ . . . . &c, ! bm VA — a/ 2 > 3 A/6 _V 42 + -y/i8 attz+zan—mbl-ba”— &c. Vi — Vd - 4 m When the fquare root of a furd is required, it may —K~an-—rn,l—an—& z±zb—I be found nearly by extruding the root of a rational quantity that approximates to its value. Thus to find the fquare root of 3+2^2, we firlt odculate ^2=1, 41421; an * nl * * =t=bmn and therefore 3+2v/,2=j, 82842, whofe root is found The fign of bm is pofitive only when rn is an odd n to be nearly 2, 41421 : So that ^3 +2^/2 is nearly ber, and the binomial propofed is am--bl. 2,41421. . But fometimes we maybe able to^ exprefs^ If any binomial^ furd is propofed whofe two numbers the roots of furds exadtly by other furds; as in this*"exhave dilferent indices, let thefe be m and /, and take n ampie the fqliare root 0f 34-2^2 forT+V2X equal to the leaft integer number that is meafure&l by m j ’ ‘V and by —/ ; and an—™-^an— zmb!-~-an—3 mb' h=fza»~-4>n IetInus order know u ofetothat x when and how this may be found, . a compound, furd, r, which , •, multi, •. will bef xPPJ-(-> J+2.yy .+y a binomial furd, whofe bV, &c. lhall give • If isx andy are quadratic furdsfquare then plied by the propofed a>n=tzbl fliall give a rational wiH be rational, and 2xy irrational; fo that 2xy prodmS. Thus j~a—l/l being given, fuppofe ^11 always be lefathan x^+yS.becaufe the difference T and — =4, therefore you have «=3, 3 and is x^+y^-s^x-y which is always pofitive Sup/ 11 ,, Ani u pole that a propoied lurd confiding of a rational part JA, i A with this, then x + and an™ irrationalA part ^B, coincides 2n-**mbl+an-'™b --« — b +&c.=a —'*- y*=A ^ was faid of o an—mJl r and x;- —A B : Therefore “by what 1 li llb >rai li 6 al b+a*~^b^--a c=,!~ 'J ~,IXbl-a b!/ v=:ai [:+a*b quations. Chap. 12th, 3 / i 1 +«TG+^+^T+^4 a /«/ +'rt s A/^+V «/ 3Xv /d‘ + y —A—X ^:—and therefore, 1, 1 = «+ ' XV ^-i-«V ' Xv ^+ a r ^ E , B q.• 3 3 ® —— and —Ax*1-}-—zi! ab-^-bifaX*/b--bX./b** which multiplied by the /a— ,4 4 ’ frorn whence we have and y* =r t/b gives an—bfn =a'-b ^ By thefe theorems any binomial furd whatfoever be- A—V^A3—B'. rp, f. ing given, you may find a furd which multiplied by it 2 1 ^erefore when a quantity partly lhall give a rational proctucl. rational, partly irrational, is propofed to have its root* Suppofe that a binomial furd was to be divided by an- extra&ed, sail the rational part A, the irrational B other, as ^20+^12, by ^5 - ^>3, the quotient may °f ths root f^11 ^ be exprefi’ed by But it may be ex- —— , and the fquare 'of the defer part/hall prefled in a more limple form by multiplying both nume- j}g A—-y A1—B ’• And as often as the fquare root of rator and denominator by that furd 'which, multiplied pofed binomial furd may be exprelfed itfelf as a biy/ 20+-y/ 12 _ a/ 20 + f 12 f 5 4- y/g/ _ nomial lurd. For example, if 34-24/2 is propoVs—Vb a/5—^/3 a/ 5 + A 3 fed, then A=3, B-2f22 and A2 —B1=9 8=x. y/ IOO + 2 y/do +6 16 4-24/60 g_p.2y/i 5 2 u t _ S 3 . ., Therefore xi — ^-f-y^A B* _ ^ In general, when any quantity is divided by a bmo— * * / — 2 mial furd, as amz±zb!, where m and / reprefent any fraAions whatfoever, take n the leaf integer number ^ — r* Therefore x4-y=i4-y/2. that is meefired by m ana —, multiply loth nutnerator To find the fquare rootaof—i-}-ydg} fuppofe2 A — ’ and denominator by att—m 4. an —f^bH-a0 — &c. .—i,B=y/—8, fo that A B*—0 and ^jlA/A —B* and the denominator of the produd n ;// become rational, ' 2 and equal to an — b —; then divide all the members of — f— — I, and ^—T"1 3 the numerator by this ratio,id quantity, and the quote therefore tlie root required is i+y/
 * 'V