48 A Treatise of Algebra.

From whence it appears that a 2 -j- o — x z , or a 1 — **will measure both a+ — x' and <z 3 — a z x —> ax 2 -}- x 3 ;Therefore, by dividing thereby, the Fraction proposeda 2 -|- x z

will be reduced to -.

Let the Fraction

a — x

ax z -\- bx 2 —- 2a z x-\-bax—2 ba z

x z — bx-\- 2 ax — labbe given. Then I first divide the Numerator by theDenominator, thus

x z 4 - 2a \ , \ *3 4 - a ? , —2a 2 \ , s — a

- b \ X ~ 2ab ) +b\* + ba\ x - 2 ba \ x +2b

| -V" — 2abx

a / ,— ia. z \2b\ Xl ' - '

Here the Remainder is

+2bS* + Z abS-'~ 2baZ

+2 a b\ xl —la z l x + 2a * b

__ -2 b z S

— 2 ab |

-J -2 b z

—4 orb

*

which divided by it’s greatest simple Divisor, ab, gives| x ~ ° r b — ^ — ax 2a; but this,

I perceive, is still further divisible, by the compoundQuantity b — a, the Quotient being x -|- 2 a. I there-fore, next, divide by *-f- 2a (instead of \ x 20b)and the Work will stand, thus

*-f- 2 a^* j.v— zab^x—b

x z -f- 2ax

— bx — 2 ab

— bx — 2 ab

Hence it appears, thatr-f- 2a will exactly measure boththe Numerator and Denominator ; whereby the Fraction

j , x z — ax-{• bx—abproposed, is reduced to - — -

.r — b

Alfc.