As the exponents of a thus 1 - decrease, and at 
fcjie same time those of b increase, “ the sum of 
their exponents is always the same, and is equal 
to the exponent of the power required.” Thus 
in the Gth power of a -{- b, viz. a 6 -f~ Ga b -j- 
■1 SaW -4- 20 a'V 4- 1 5a 2 b’ 4- Gab' 3 4- b\ the ex- 
ponents of a decrease in this order, 6, 5, 4, 3, 2, 
з, 0; and those of b increase in the contrary or- 
der, 0, 1,2, 3, 4, 5, 6. And the sum of their ex- 
ponents in any term is always 6. 
To find the co-efficient of any term, the co- 
efficient of the preceding term being known ; 
you are to “ divide the co-efficient of the pre- 
ceding term by the exponent of b in the given 
term, and to multiply the quotient by the ex- 
ponent of a in die same term., increased by unit.” 
Thus to find the co-efficients of the terms of the 
Oth power of a -|- b, you find the terms are 
o'\ ab, aV/. U : L\ a 2 b\ ab', b ; and you kno\y 
the co efficient of the first term is unit, there- 
fore, according to the Rule, the co-cfficieat of 
the second term will be 
-y- X 5 4-1=6; 
that of the third term will be 
S— x 4 4- 1 3 X 5 = 15; 
2 1 
that of the fourth term will be 
15 — — 
■y X 3 4~ 1 = 5 X 4, 
of the following terms will b 
able to the preceding table. 
In general, if a -}- b is to be raised to any 
power m, the terms, without their co-efficients, 
will be, a m , a” 1 ~ 1 b, a m ~ 2 b 2 , a’ n ~ 3 Z> ! , a m — V, 
и , n — -b', Si c. continued till the exponent of b 
becomes equal to m. 
The co-efficients of the respective terms, ac- 
cording to the last Rule, will be 
m — ■ 1 w — 1 m — 2 
I > m, m X — =-> m X — 
: 20 ; and those 
15, 6, 1, agree- 
m X 
m X — 
- 1 m — 
X — - 
1 
X 
2 3 
m — 3 
X 
m — 3 m — 4 
X 
3 4 
&c. continued until you have one co-efficient 
more than there are units in m. 
It follows therefore by these last Rules, that 
a 4~ b?)" ! — a m 4~ tna m — 'b m X — ■ — — X 
■ 2 b 2 4 . 
m — 1 
m X 
— 1 
A L G B B H A. 
denominates the root required. Thus, the square 
root of a 8 is = a 4 ; and the square root of 
a 2 b 3 c 2 is a 2 b ' C. The cubfc root of a 1 ' b'hr. E. I?., 
— a 2 b\ and the cube root of .y 9 y b z“ is .v'y 
The ground of this rule is obvious from the rule 
for Involution. The powers of any root are 
found by multiplying its exponent by the index 
that denominates the power ; and therefore, 
when any power is given, the root must be 
found by dividing tl-.e exponent of the given 
power by the number that denominates the 
kind of root that is required. 
It appears from what was said of Involution, 
that ‘‘ any power that has a positive sign, may 
have either a positive or negative root, if the 
root is denominated by any even number. 
Thus the square root of -j- «' may be -{- a, or 
— a, because 4 * n X 4“ <! > or — a X “ 8 ives 
-j- a 1 for the product. 
But if a power have a negative sign, “ no 
root of it denominated by an even number can 
be assigned,” since there is no quantity that mul- 
tiplied into itself an even number of times can 
give a negative product. Thus the square root 
of — a 2 cannot be assigned, and is what we call 
an “ impossible or imaginary quantity.” 
But if the root to be extracted is denominated 
by an odd number, then shall the sign of the 
root be the same as the sign of the. given num- 
ber whose root is required. Thus the cube root 
of —a 2 is — a, and the cube root of — af'E 2 is 
— a l h. 
If the number that denominates the root re- 
quired is a divisor of the exponent of the given 
power, then shall the root be only a “ lower 
power of the same quantity.” As the cube root 
of a 12 is u 4 , the number 3 that denominates the 
cube root being a divisor of 12. 
But if the number that denominates what sort 
of root is required is not a divisor of the expo- 
nent of the given power, “ then the root re- 
quired, shall have a fraction for its exponent.” 
Thus the square root of a 1 is al . ; the cube root 
of a 3 is and the square root of a itself is a\. 
These powers that have fractional exponents 
are called “ imperfect powers or surds and are 
otherwise expressed by placing the given power 
within the radical sign y , and placing above 
the radical sign the number that denominates 
what kind of root is required. Thus 
53 
cond member of the root. Add this second 
member to the double of the first, and multiply 
their sum (2 a -j- b) by the second member b, and 
subtract the product (2 ah 4 - br) from the fore- 
said remainder (2 ab -j- 5 2 ), and if nothing re- 
mains, then the square root is obtained ; and in 
this example it is found to be a -|- b. 
The manner of the operation is thus ; 
a 2 -f 2 ab - b 2 (a b 
la -j- b 
X l> 
\ 2 ab b 2 
) ‘■lab -j- b 2 
0 
But if there had been a remainder, you must 
have divided it by the double of the sum of the 
two parts already found, and the quotient would 
have given the third member of the root. ■ 
Thus, if the quantity proposed had hp§n 
a 2 -j- 2 ab lac -(- b 2 -}- 2ie -f- r 2 , after procwxl-- 
jng as above, you would have found theTre- 
mainder lac -j- 2 be -J- <4 which divided ijy 
la -f lb gives c to be annexed to a b as the 
third member of the root. Then adding c to 
la -J- lb, and multiplying their sum la -}- 25 
-j- c by c, subtract the product lac -j- Ibc -j- c 2 
from the foresaid remainder : and since nothing 
now remains, you conclude that a 4~ 5 -j- is 
the square root required. 
The operation is thus : 
a 2 -f lab -j- lac -}- b 2 -|- 2 be -j- c 2 (u.-J- 5-{- c 
X 
T“ x 
■ 3 5 ’ 4 - 
m X — — X — X — — X ~ 4 £ 4 4~> 
&c. which is the general Theorem for raising a 
quantity consisting of two terms to any power m. 
If a quantity consisting of three, or more 
terms is to be involved, “ you may distinguish it 
into two parts, considering it as a Binomial, and 
raise it to any power by the preceding Rules ; 
.and then by the same rules you may substitute 
instead of the powers of these compound parts 
their values.” Thus, 
a 4 * b 4 - 
And a -f 
a — | — o —j — c a — b 4* 1° 
lab -M 2 4- lac + 2 be + ‘ 2 
4- c = a 4~ 
b‘ 4- 3 \c x a 4" b 4* 
3 a 2 b 4- 3 ab 2 -j- 5 3 4~ 
3 c 2 X a 4 * b — 
Sa 2 c 4 - 6abc -j- 3 Pc -f- Zac 2 -]- 3 be 1 4 - c\ 
In these examples, a -|-54. r, is considered 
as composed of the compound part a b and 
the simple part c ; and then the powers of a-\- b 
are formed by the preceding rules, and substi- 
tuted for a 4 - b 2 and a -j- b L% 
\/ a\ «y=\ // a''; and a n 
In 
numbers the square root of 2 is expressed by 
4/ 2 , and the cube root of 4 by \/ 4. 
These imperfect powers or surds are “ multi 
plied and divided, as other powers, by adding 
and subtracting their exponents.” Thus, 
ai X al. = flk = 
X *5 — a r H" 4 — a T% — 
« r. 
3 . 2 
They arc involved likewise, and evolved after 
the same manner as perfect powers. 
6 x 1 
square of ajL is a 2 — - , 
Thus the 
’ 2 = a 3 ; the cube of 
X 
cc ^ The square root of a\~ is a X 
— aj, the cube root of aj is aj. 
The square root of any compound quantity 
as a 2 4- lab 4 - b 2 is discovered after this man- 
ner. “ First, take care to dispose the terms ac- 
cording to the dimensions of the alphabet, as in 
Division ; then find the square root of the first 
term aa, which gives a for the first member of 
The reverse of Involution, or the resolving ■ the root. Then subtract its square from the 
of powers into their roots, is called Evolution. | proposed quantity, and divide the first term of 
The roots of single quantities are easily extracted the remainder (lab -}- by tl ie double of that 
by dividing their exponents by the number that ■ member, vjz. 2a, and the quotient It is ffie se- 
Of Evolution. 
2 a 4 _£\ lab 4 - la 
X t>) lab 
+ * + 
4- 4 
>bc 4 -, 
lb 4 - c\ lac -\ 
|- 2 be -| 
_ 2 
X e) lac - 
f- Ibc -j 
- 5“ 
0 
. 0 
0 
Another Example. 
xx — ax 4 " \ aa ( 
v — 4‘ i 
lx — 
X — 
t) - 
— ax 4 - 4^ 
X- 4- \a 
0 . o 
In general, to extract any root out of any 
given quantity, “ First range that quantity ac- 
cording to the dimensions of its letters, and ex- 
tract the sa;d root out of the first term, and that 
shall be the first member of the root required. 
Then raise this root to a dimension lower by 
unit than the number that denominates the root 
required, and multiply the power that arises by 
that number itself ; divide the second term of 
the given quantity by the product, and the quo- 
tient shall give the second member of the root 
required.” 
Thus, to extract the root of the 5th power 
out of a 2 -]- 5« 4 5 -j- lOa’b 2 -[- 1 0 a 2 b : 4 - 5ab * 4 - b\ 
I find that the root of the 5th power out of a h 
gives a , which I raise to the 4tli power, and 
multiplying by 5, the product is 5a 4 ; then di- 
viding the second term of the given quantity 
5a 4 b by 5a 4 , 1 find b to be the second member ; 
and raising a -{- b to the 5th power, and sub- 
tracting it, there being no remainder, 1 Con- 
clude that a 4-5 is the root required. If the 
root has three members, the third is found after 
the same manner from the first two considered 
as one member, as the second member was 
found from the first ; which may be easily un- 
derstood from what was said of extracting the 
square root. 
In extracting roots it will often happen that 
the exact root cannot be found in finite terms j 
thus the square root of a 1 -J- x 2 is found to be 
5x 8 
a + 
+ 
1 6« 3 
128 « 
7 4~j & c * 
