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 II A. 2 oo A L O 1 But though x and y are not quadratic furds or roots “ t, fo fliall the root required be —^~t~——, i^of integers, if they are the roots of like furds, as if they are equal to and where m and n are “ the c root of A=!=B can be extraCled. Examp. I. Thus to find the cube root of /qhS integers, then A = #-'-ir«*Vz an(l j-B~ -F25, Vre have A*—Bi=343-, whofe divifors are 7, 7, 7, 5 whence «=7, and Q=i. Further, A-f-BX/Q^ that A —Xz and x* = 5 is a little more than 56, whofe neareft «~r» Xz4-w—«j^Xz _ : jn*/z, y - A—a/A2—B1 is,cube/968+2 root is 4. Wherefore rz=4. Again,-dividing 4/968 z /, by its greatell rational divifor, we have A 4/ Q^=22/2, Ay/z, and x+y = /r«v' .+ 'V «v z- The part A here eafily diftirguUhes itfelf from B by its being greater, and the radical part 4/2=/; and r+ T~- or —, in if x and_y are equal to a/nuyz thenx*+ the neareft integers, is 2 —t. And laftly, ts — 2^/2, ,/;;; / and Wy'z-|-«y//-(-2V' ' V 2/^ So that if z or / —«=i, and X/Q_= v' 1=1. Whence2/2+j be not multiples one of the other, or of fome number is the root, whofe cube, upon trial, I find to be 4/968 that meafures them both by a fquare number, then will + 25A jtfelf be a binomial. Examp. i Let7l x--y--z exprefs any trinomial furd, its fquare x*-{xII. To find the cube root of 68—4/4374, y --z -t-2xy+2xz-{-2yz may be fuppofed equal to A+B we have A —whofe divifors are 5, 5, 5, 2. as before. l3ut rather multiply any two radicals as nx/ by zxz, and1 divide by the third zyz which gives the Thence »=5X2=iO, and Q34, and 4/A+BX4/Q^ °r quotient 2X rational, and double the fquare of the furd /68+4/4J74X2 is nearly 7-r; again A4/Q^ or 68X x required. 2 The2 fame rule ferves when there are four quantities x +ry -hzi+rx+2xj’d-2XJ+2xz-i-2jz-{-2^/d- 4/4=13 6X/1, that is, j=i, and;' + or —' !zjy multiply 2xv by ax/, and the produft qx^sy divided is nearly =4=/. Therefore ts—q, —/;=4/6, by 2sy gives 2x° ^ rational quotient, half the fquare of 2x. In like manner zxy'X.zyz—qy'xz, which divided by and 4/0^4/4—-'/2, whence the root to be tried is axz another member gives 2>*, a rational quote, the 4-/6 half of the fquare of 2>. In the fame manner z and s may be found; and -their fum x~-y--z--j, the fquare root of the feptinomial x*+g'2+za-i--fz+25y+2k/-|-2xz-H 2_yz+2>/, difcoveved. XIV. Of the Genesis ffwa? ResoluFor example, to find the fquare root of ib+v^24+ Chap. tion of in general; and the ^■40+4/60; I try which I find to be number of Roots an Equation of any Degree ^/i6=4, the half of the fquare root of the double of may have. which, viz. 4Xy'3=V'2, is one member of the fquare After the fame manner, as the higher powers are root required; next V4° — 6, the half of the produced by the multiplication of the lower powers of fame root, equations of fuperior orders are generafquare root of the double of which is V 3 another mem- the ted by the multiplication of equations of inferior orders ber of the root required; laflly, — =10, which involving the fame unknown quantity. And “ an equaet any dimenfion' may be confidered as produced gives 4/5 for the third member of the root required : tiontheof multiplication of as many firople- equations as from which we conclude, that the fquare root of 10+ ““ itby has dimenfions, of any other equations whatV24+-1/40+60. is /2+/3+/S 5 and trying, you “ foever, if the fum ofortheir is equal to the find it fucceeds, fince multiplied by itfelf it gives the pro- “ dimenfion of that equation.”dimenfions Thus,’ any cubic equapofed quadrinomial. may be conceived as generated by the multiplication For extracting the higher roots of a binomial, whofe tion of three fimple equations, or of one quadratic and one two members being fqnated are commenfurable numbers, fimple equation. A biquadratic is generated by the there is the following multiplication of four jimpte equations, or of /wo quaRule.' “ Let the quantity be Az±=B, whereof A is dratic equations, or, laftly, of one cubic and one fimple “ the greater-part, and -r the exponent of the root equation. “ required. Seek the lead number n whdfe power If the equations which you fuppofe multiplied by one “ -nc is divifible by AA—BB, the quotient being another are the fame, then the equation generated will elfeisbut fomeiia)t,luti-n; power of thofe equations, and Compute / A+B X-v/m the- neareft integer bethenothing merely of which we have “ number, which fnppofe to be r. Divide A/Q_by operation already : arm, when any fuch equation is given, “M its greateft rational divi’for, and Jet the quotient be /, treated the fimple equation by whofe multiplication it is produ,by evolution, or the extraction of a root. and let f + ~T in the'hearefi: integer number, be cedButis found wfien the equations that ■ are fitppeftd to betiplied mul-