ALGEBRA. 
Ex. 3. To extract the square root of 1 — |— *. 
1 + * G + f — | &c - 
1 
*+'!)“ 
*+*-?) 
It appears from the second example, that 
a trimonial a 2 — a x -| — - in yvhich four 
4 
times the product of the first and last terms 
is equal to the square of the middle term, is 
r 2 
a complete square, or a 2 x — X^ = fl 2 x 2 . 
4 
The method of extracting the cube root is 
discovered in the same manner. 
fl 3 3 a 2 b — |— 3 a b 2 — j— b 3 ( a — |— b 
a 3 
4 
x 2 a? 5 . x 4 
4 * ~ 8 " "^64 
.r 3 a? 4 _ 
¥“ 64 &C - 
3 a 2 ) 3 a 2 A -[- 3 a b 2 4 A 3 
3 a 2 b -j- 3 a b 2 -j- b 3 
—— * 
The cube root of «4 3 a 2 b 4 3 a b 2 - f- A 3 is a 
-}- b ; and to obtain a b from this com- 
pound quantity, arrange the terms as before, 
and the cube root of the first term, a 3 , is a, 
the first factor in the root ; subtract its cube 
from the whole quantity, and divide the first 
term of the remainder by 3 a 2 , the result is b, 
the second factor in the root; then subtract 
S a 2 A -)- 3 a h 2 -j- A 3 from the remainder, and 
the whole cube of <z-|-A has been subtracted. 
Jf any quantity be left, proceed with a -j- A 
as a new a, and divide the last remainder by 
5 . a — 6~1 2 for a third factor in the root ; and 
thus any number of factors may be obtained. 
ON SIMPLE EQUATIONS. 
If one quantity be equal to another, or to 
nothing, and this equality be expressed alge- 
braically, it constitutes an equation. Thus, 
x — a = A — x is an equation, of which .r 
— a forms one side, and A — x the other. 
When an equation is cleared of fractions 
and surds, if it contain the first power only of 
an unknown quantity, it is called a simple 
equation, or an equation of one dimension : if 
the square of the unknown quantity be in any 
term, it is called a quadratic, or an equation 
of two dimensions ; and in general, if the 
index of the highest power of the unknown 
quantity be n, it is called an equation of a 
dimensions. 
In any equation, quantities may be trans- 
posed from one side to the other, if their signs 
be changed, and the two sides will still be equal. 
Let *-(-10 = 15, then by subtracting 10 
from each side, *-{-10 — 10 = 15 — 10, or 
*=15— 10. 
Let * — 4 = 6, by adding 4 to each side, 
x — 4 -{- 4 = 6 -(- 4, or ,r = 6 -|- 4. 
If* — a -J- A — y ; adding a — A to each 
side, * — a -(- A -j- a — A = y -j- a — - A ; or 
* = y -{- a — A. 
Hence, if the signs of all the terms on 
each side be changed, the two sides will still 
be equal. 
Let x — a = A — 2 * ,• by transposition, 
— A-j-2.r = — x-\-a ; or a — * = 2* — A. 
If every term, on each side, be multiplied by 
the same quantity, the results will be equal. 
An equation may be cleared of fractions, 
by multiplying every term, successively, by 
the denominators, of those fractions, except- 
ing those terms in which the denominators 
are found. 
5 x 
Let 3 ,r -}- — 34 ; multiplying by 4, 
12*-f5* = 136, or IT *=136. 
If each side of an equation be divided by the 
same quantity, the results will be equal. 
Let 1 T * = 136 : then * = =' 8. 
IT 
If each side of an equation be raised to the 
same power, the results will be equal. 
Let *2 = 9; then * = 9 x 9 = 81. 
Also, if the same root be extracted on both 
sides, the results will be equal. 
Let * = 81 ; then *2 = 9. 
To find the value of an unknown quantity in 
a simple equation. 
Let the equation first be cleared of frac- 
tions, then transpose all the terms which in- 
volve the unknown quantity to one side of 
the equation, and the known quantities to 
the other ; divide both sides by the co-effi- 
cient, or sum of the co-efficients, of the un- 
known quantity, and the value required is 
obtained. 
u Ex. 1. To find the value of * in the equa- 
tion 3 * — 5 = 23 — *. 
by transp. 3*4 * = 23 4-5 
or 4 * = 28 
.... 28 
* by division * = — = T, 
