ALGEBRA. 
Ex. 5. 
* — a)x 3 — px 2 -\-qx — J'(x 2 -|-u — p.x-\-a* — pu-\-q 
o' 3 — ax 2 
a—p. x 2 -±-q.v 
a — p.x 2 — a 2 — pa.zc 
-j-a' — pa-\-q.x-—r 
a 1 — pa-\-q,x — a 3 — pa 2 -\-qa 
Remainder d — pa 2 -\-qa — r 
ON THE TRANSFORMATION OF FRACTIONS 
TO OTHERS OF EQUAL VALUE. 
If the signs of all the terms both in the nu- 
merator and denominator of a fraction be 
changed, its. value will not be altered. For 
- a h 
-b=P±; 
~h a 
and 
a h 
a 
If the numerator and denominator of a frac- 
tioR.be both multiplied, or both divided by the 
same quantity, its value is not altered. For 
a c a ^ a r y z ,r y 
be b’ a b c z be 
Hence, a fraction is reduced to its lowest 
terms, by dividing both the numerator and 
denominator by the greatest quantity that 
measures them both. 
= i. e. a is contain- 
ed in x y, m-\- n times, or it measures 
x i V by the units in 
Now it appears from what has been said, 
that a — p b~c, and b — q c — d ; every 
quantity therefore which measures a and b, 
measures p b, and a —p b, or c ; hence also it 
measures q c, and b — q c, or d ; that is, every 
common measure of a and b measures d. 
Ex. To find the greatest common measure 
of a 4 — x 4 and o 3 — a 1 x — a x 2 -j- x 3 , and to 
( 1‘ ^ ry>4 
reduce ; — — : to its lowest 
3 — a 2 .r — a x 2 x : 
terms. 
a 3 — a 2 x — a x 2 -|- x 3 ) a 4 — x 4 (a -}- .r 
a 4 — a 3 x — a 2 x 2 -\-ax 2 
o 3 x-J-a i x 2 — ax 3 — x 4 
a 3 x — -a 2 ,v 2 — ax 3 -^-x 4 
2a 2 , r 2 - — 2x 4 
leaving out 2 x 2 , which is found in each term 
of the remainder, the next divisor is a 2 — x 2 . 
a 2 — x 2 ) a 3 — a 2 x — a x 2 x s ( a — x 
a 3 — a x 2 
— a 2 x -\-x 2 
— a 2 r-f-r* 
* 
The greatest common measure 'of two quanti- 
ties is found by arranging them according to 
the powers of some letter, and then dividing the 
greater by the less, and the preceding divisor al- 
ways by the last remainder, till the remainder is 
nothing ; the last divisor is the greatest common 
measure required. 
Let a and b be the two quan- 
tities, and let i be contained in 
a, p times, with a remainderc; 
again, let c be contained in b, 
q times with a remainder d, 
and so on, till nothing remains; 
let d he the last divisor, and it 
will be the greatest, common 
measure of a and b. 
b)a(p 
c) b ( q 
d)c ( r 
0 
The truth of this rule depends upon these 
two principles ; 
1. If otie quantity measure another, it will 
also measure any multiple of that quantity. 
Let x measure y by the units in n, then it 
will measure cy by the units in n c. 
2. If a quantity measure two others, it will 
measure their sum or difference. Let a be 
contained in x, m times, and in y, n times; 
then ma — x and na = y ; therefore x -1- y 
dr — x z is therefore the greatest common 
measure of the two quantities, and if they be 
respectively divided by it, the fraction is re- 
duced to 1 , its lowest terms. 
a — x 
The quantity 2 x 2 , found in every term of 
one of the divisors, 2 a 2 x 2 — 2 x 4 , but not in 
every term of the dividend, a 3 — a 2 x — a x 2 
+ x 3 , must be left out; otherwise the quo- 
tient will be fractional, which is contrary to 
the supposition made in the proof of the rule; 
and by omitting this part, 2 x 2 , no common 
measure of the divisor and dividend is left 
out; because, by the supposition, no part of 
2 x 2 is found in all the terms of the dividend. 
To find the greatest common measure of 
three quantities ab c ; take d the greatest 
common measure of a and b, and the great- 
est measure of d and c is the greatest com- 
mon measure required. In the same man- 
ner, the greatest common measure of four or 
more quantities may be found. 
If one number be divided by another, and 
tlie preceding divisor by the remainder, ac- 
cording to what has been said, the remain- 
der will at length be less than any quantity 
that can be assigned. 
