A Treatise os Algebra.

0,1 C* st 1

Likewise if --— — —, then you will

a 2 — x 1 za x x

have a' xza-\- x y.x — c 2 X a 2 — * 2 X * ==: a* x a 2 —

X 2« -f- ar or -j- a*x 2 — aVr -f* r 2 *3 — 2a J -f- « 4 x

— 2a J * 2 — a 2 * J . Lastly, if x-\- s/a 1 + ** = -7^=.

_ v a 2 4-ar 2

there will come out 2Xv / fl 2 +* 2 + « 1 + * I = 2 a 2 .

5 0 . Also, if in your Equation there be an irreducible Surd ,wherein the unknown Quantity is concerned , all the otherTerms must be transposed to the contrary Side, with theirSigns changed , and both Sides involved to the second , thirder fourth Power , &c. according as the Root is a Square ,Cube or Biquadratic One: And, if there be more suchSurds than one, the Operation is to be repeated. Thus,if you have ar x v^a 2 -f- x 2 a 2 -}- * 2 2 a 1 , by trans-

posing, or subtracting a 2 + ar 2 from each Side, you willhave x X fa 2 -{- x 2 — a 2 — ar 2 , which being in volved tothe second Power, (or squared) there arises ar 2 xa*-f-x*

— a+—aa 2 ar 2 -|- a 4 , or a 2 * 2 -f- *4 z= a 4 — 2a 2 ar*-f- * 4 ,

6 °. Having by the two preceding Rules (if need be) clear-ed your Equation from Fractions and irreducible Surdsy andordered the several Terms thereof according to Rule the third,let the whole Equation be divided by tire Coefficient (or theSum of the Coefficients) of the highest Power of the unknownQuantity: And then if your Equation be a simple one,the Work is at an End - but if Quadratic, or Cubic,isfc. that is, if the highest Power of the unknown Quan-tity rises to two, three, or more Dimensions, somethingfurther is still to be done, and recourse must be had tothe particular Methods for resolving these kind of Equa-tions ; which shall be hereafter considered in a proper Place.

I shall here subjoin some Examples of reducing Equa-tions, wherein the foregoing Rules promiscuously ob-tain.

EXAMPLE I.

Let 1 o 4 - — — = t 6 ; to find x.

12 —x

Here,