54 
The operation is tl us :• 
+**(«+ 4 - ■£+ 
: — ,&C. 
16a 
~ a ^Ta) 
2a 1 
+ 
■la 2 
+ 
8a’ 
8a 4 
64a 5 
1 8a 4 64a 5 
After the same manner, the cube root of 
<^ + a 3 will be found to be 
+ + & c . 
^ 3a 2 9a 5 ^ 8 la 8 243a 11 
« The general theorem which we gave for the 
involution of binomials, will serve also for their 
evolution ;” because, to extract any root of a 
given quantity, is the same thing as to raise that 
quantity to a power whose exponent is a frac- 
tion that has its denominator equal to the num- 
ber that expresses what kind of root is to be ex- 
tracted. Thus, to extract the square root of 
« -j- b is to raise a -{- b to a power whose expo- 
nent is •§. Now, since a -j- b —a -X. m x 
... m — l ... , 77i — 1 
a m — 1 5 - j- ?7i X — - — - a ' n ~ 2 b~ 4- m X — — — 
x Z _ X a m ~ *b\ &c. 
3 
§, you will find 
1 X a “ * b + f X - { 
“ J X. — i a 2 b\ Sec. 
T , b b 2 b 2 
- a i + — r ~ 0.4 + -T7T-T ~ > &c - 
3 
Supposing m - 
4~ b T — a \ -} 
fi X 
2 a\ Sal 1 16a5 
1 2 2 
And after this manner you will find that 
l v -2 v 4 v& 
4" 
a O . 
Sa 2 
lCn' 
& c. as 
ALGEBRA. 
and third, and from their sum subtract the first 
term, the remainder shall give the fourth arith- 
metical proportional required.” 
In a series of arithmetical proportionals, “ the 
sum of the first and last terms is equal to the 
sum of any two terms equally distant from the 
extremes.” If the first terms are a, a 4- b, 
a -j- 2b, Sec. and the last term .v, the Jast term 
blit one will be .v — b, the last but two a- — 2b, 
the last but three x — 3 b. See. So that the first 
half of the terms, having those that are equally 
distant from the last term set under them, will 
stand thus : 
a, a -|- b, a -J- 2 b, a -|- 3 b, a -j- 4 b, &C. 
x , x — b T x — 2b, x — 3b, x — 4b, 
a -j- x, a — J— x, a -|* x, a -J— A", a -1- .v, &C. 
And it is plain that if each term be added to 
the term above it, the sum will be a -)- x, equal 
to the surh of the first term a, and the last term 
x. From which it is plain, that “ the sum of all 
the terms of an arithmetical progression is equal 
to the sum of the first and last taken half as 
often as there are terms,” that is, the sum of an 
arithmetical progression is equal to the sum of 
the first and last terms multiplied by half the 
number of terms. Thus, in the preceding se- 
ries, if n be the number of terms, the sum of all 
the terms will be a -j- .v x — • 
The common difference of the terms being b, 
and b not being found in the first term, it is 
plain that “ its co-efficient in any term will be 
equal to the number of terms that precede that 
term.” Therefore, in the last term x, you must 
have n — 1 x b, so that a- must be equal to 
a n — 1 X b. And the sum of all the terms 
being a 4- x X ~, it will also be equal to 
2 an -{- n l b — nb nb — b 
, or to a 4 X ». 1 hus, 
2 1 2 
for example, the series 1 -]~ 2 -j- 3 -]- 4 -f- 5, &c. 
continued to a hundred, must be equal to 
2 X 100-f 10000 — 100 
5050. 
Of Proportion, 
When quantities of the same kind are com- 
pared, it may be considered either how much 
the one is greater than the other, and what is 
their difference ; or, it may be considered how 
many times the one is contained in the other, 
or, more generally, what is their quotient. The 
first relation of quantities is expressed by their 
arithmetical ratio; the second by their geome- 
trical ratio. That term whose ratio is enquired 
into is called the Antecedent, and that with 
which it is compared is called the Consequent. 
When of four quantities, the difference be- 
twixt the first and second is equal to the differ- 
ence betwixt the third and fourth, those quan- 
tities are called Arithmetical Proportionals ; as 
the numbers 8, 7, 12, 16 ; and tbe quantities 
a, a -\- b, e, t -\- b. But quantities form a series 
in Arithmetical Proportion, when they “ in- 
crease or decrease by the same constant differ- 
ence ;” as these, a, a -j- b, a -j- 2b, a -}- 3b, 
“ -j- 4b, Sec. x, x — b, x — 2b, Sec. or the num- 
bers 1, 2, 3, 4, 5, &c. and 10, 7, 4, 1 , — 2, —5, 
— 8, Sec. 
In four quantities arithmetically proportiona 1 , 
“ the sum of the extremes is equal to the sum ot 
the mean terms.” Thus, a, a -f- b, e , e -f- b, 
ure arithmetical proportionals, and the sum of 
the extremes (,. e -(- 5) is equal to the sum of 
the mean terms (a -}~ b -j- e). Hence, to find 
the fourth quantity arithmetically proportional 
to any three given quantities ; “ add the second 
If a series have (0) nothing for its first term, 
then “ its sum shall be equal to half the product 
of the last term multiplied by the number of 
terms,” For then, a being — O, the sum of the 
terras, which is in general a x X \ > will in 
this case be 
From which it is evident, that 
“ the sum of any number of arithmetical pro- 
portionals beginning from nothing, is equal to 
half the sum of as many terms each equal to the 
greatest term.” Thus, 
0 — |— 1 — {— 3 — {— 4 — j— A — J— 6 — j— 7 — {— S — 9 
9 -f 9 -f- 9 -f 9 + 94-9 -|~ 9 -f 9 -f 9 + 9 
— 10 x 9 _ 45 
2 
“ If of four quantities the quotient of the first 
and .second be equal to the quotient of the third 
and fourth, then those quantities are said to be 
in geometrical proportion.” Such are the num- 
bers 2, 6, 4, 12; and the quantities a, ar, b, hr ; 
which are expressed after this manner ; 
2 : 6 :: 4 ; 12 . 
a \ ar \ \ b * br. 
And you read them by saying, As 2 is to 6, so 
is 4 to 12 ; or as a is to ar, so is b to br. 
In four quantities geometrically proportional, 
“ the product of the extremes is equal to the 
product of the middle terms.” Thus, a x br — 
ar x b. And, if it is required to find a fourth 
proportional to any three given quantities, 
“ multiply the second by the third, and divide 
their product by the first, the quotient shall give 
the fourth proportional required.” Tims, to 
find a fourth proportional to a, ar, and b, ] mul- 
tiply ar bv b, and divide the product arb by the 
first term’ a, the quotient hr is the fourth pro- 
portional required. 
When a series of quantities increase by one 
common multiplicator, or decrease by one com- 
mon divisor, they are said to be in “ geometrical 
proportion continued.” 
As, a, ar, a< 2 , ar 2 , a> A , ar'’, Sec. or, 
a a a a a 
a i > i T» T> ~ &c. 
r r r r r * 
The common multiplier or divisor is called 
their “ common ratio.” 
In such a series, “ the product of the first and 
last is always equal to the product of the second 
and last but one, or to the product of any two 
terms equally remote from the extremes.” In the 
series a, ar, ar 2 , ar 3 , S< c. if y be the- last term, 
then shall the four last terms of the series be 
y , - y —, ~ ; now it is plain that a x y =; 
y_ _ 
ar X 
— ar 2 X ~~T — ar X 
Sec. 
“ The sum of a series of geometrical propor- 
tionals wanting the first term, is equal to the 
sum of all but the last term multiplied by the 
common ratio.” 
For ar -j— ar* — p- ar , Sec. — | - — — j— " — |— y — ^ 
r x a -j- ar -|- ar i ^ c - 4~ Ar 4" 4* Ar 4~ ~ ' 
Therefore, if s be the sum of the series, s — a 
will be equal to s — y X r \ that is, s — a — 
, yr — a 
sr — yr, or sr — s = yr — a , ana s —*■ 
Since the exponent of r is always increasing 
from the second term, if the number of terms 
be a, in the last term its exponent will be « — 1. 
n — 1 , n — I -J-I 
yr ~ ar 
ar n • — a 
, ) = —■ So that 
having the first term of the series, the number 
of the terms, and the common ratio, you may 
easily find the sum of all the terms. 
If it is a decreasing series, whose sum is to be 
found, as of y -f- y — {- — -f- — Sec. 4- ar 2 4- 
r r r 
ar 2 -j- ar -j- a, and the number of the terms be 
supposed infinite, then shall a, the last term, be 
equal to nothing. For, because », and conse- 
quently r >l , is infinite, a = — /\ZT~ — 
Therefore y — ar ; and 
= ./;and, = (^) = 
The sum of such a series s = 
— 1 
which is a 
finite sum,' though the number of terms be infi- 
nite. Thus, 
i + i + 5 + } + A+- &c - = 5 “T = " 
•»* » •«+*+*+*+*'•=; x 3 ’ 
1 
Of Equations. 
Drfinit. I. An equation is a proposition assert- 
ing the quality of tw r o quantities, and is ex- 
pressed by placing the sign = between them. 
Defuut. 2. Equations containing only one un- 
known quantity and its powers, are divided into 
orders, according to the highest power of the 
unknown quantity to be found in any of its 
terms. 
See the Rules under the next general head. 
