30 
ALGEBRA. 
a pound is nst to be multiplied by a pound, or 
a debt by a debt, or a line by a line. We shall 
hereafter consider the analogy that there is be- 
twixt rectangles in geometry and a product of 
two factors. 
If the quantities to be multiplied are simple 
quantities, find the sign of the product by the 
last rule ; after it place the product of the -co- 
efficients, and then set down all the letters after 
one another, as in one word. 
examples. 
Mult. 4- a I — 
B y *- 1 — l) | — 4^ 
1 
6.v 
— 5 a 
Prod. ab — 8 ai 
* — 30 a x 
Mult. — 8 a 
4" 
3 ab 
By — 4a [ 
5ac 
Prod. -j - 82 ax 
— 
IS.iabc 
To multiply compound quantities, you must 
multiply every part of the multiplicand by all 
the parts of the multiplier taken one after an- 
other, and then collect all the products into one 
sum : that sum shall he the product required. 
Mult, a -J- b 
y a — j — b 
By 
Prod. 
EXAMPLES. 
2a — 36 
4 a -j- 5b 
:4- ab 
-j- ah -j- bb 
8 aa — \2ab 
-J- 10 ab — 1 56b 
Sum aa -f- 2 ab -}- bb 
Mult. 2 a — 4 b 
By 2 a -(- 4b 
8 aa 
1 5bb 
Prod. 
4 aa — 8 ab 
-j- 8 ab — I6bb 
2 ab — 
xx — ax 
x -j- a 
xxx — axx 
-j- axx — aax 
Sum 4 aa . . 0 . — 16bb xxx . . 0 . — aax 
Mult. aa ab -\- bb 
By a — b 
j C aaa -j- aab T- abb 
Pr0d - 1 -aab -abb -bib 
Sum aaa ... 0 .... 0 . — bbb 
Products that arise from the multiplication of 
two, three, or more quantities, as abc, are said 
to be of two, three, or more dimensions ; and 
those quantities are called factors or roots. 
If all the factors are equal, then these pro- 
ducts are called powers ; as aa , or aaa, are 
powers of a. Powers are expressed sometimes 
by placing above the root to the right hand a 
duce them. 
Thus, 
J 5 fist'] 
1 power of the 
- > 
2 d f 
root a, and 
3d ) 
► is shortly 
S J 
1 4th y 
. expressed 
.s ( 
_5th 
) thus, 
These figures which express the number of 
factors that produce powers, are called their 
indices or exponents; thus 2 is the index of a 2 . 
And powers of the same root are multiplied by 
adding their exponents. Thus a 2 x a 2 = a\ 
a 4 X a 1 = a 7 , a' X “ = « 4 - 
Sometimes it is useful, not actually to multi- 
ply compound quantities, but to set them down 
with the sign of multiplication (x) between 
them, drawing a line over each of the compound 
factors. Thus a -f- 6 X a — b expresses the 
product of a ,-f- b multiplied by a — b. 
Of Division. 
The same rule for the signs is to be observed 
in Division as in Multiplication ; that is, if the 
signs of the dividend and divisor are like, the 
sign of the quotient must be -j- ; if they are un- 
like, the sign of the quotient must be — . This 
will be easily deduced from the rule in Multi- 
plication, if you consider that the quotient must 
be such a quantity as, multiplied by the divisor, 
shall give the dividend. 
The general rule in Division is, to place the 
dividend above a small line, and the divisor un- 
der it, expunging any letters that may be found 
in all the quantities of the dividend and divisor, 
and dividing the co-efficients of all the terms by 
any common measure. Thus, when you divide 
\0ab -j- 1 5ac bv 20 a/.', expunging a out of all the 
terms, and dividing all the co-efficients by 5, 
i ■ . 2b 3c . 
the quotient is — — — ; and 
2b) ab -f- bb 
f 5x — 9y 
12 ab) 30 ax — 5 -lay ( — — -. 
V 2b 
/ 4b 4- 3c 
4 aa) 8 ab Gac [ — — . 
V 2 a 
[ 5a 
And 2 be) 5 abc I . 
Powers of the same root are divided by sub- 
tracting their exponents, as they are multiplied 
by adding them. Thus, if you divide a' by a 2 , 
the quotient is a s ~- 2 , or a'. And l>' divided by 
b~, gives b r> — 4 , or b 1 ; and a 7 b’' divided by a 2 b ', 
gives a' b 1 for the quotient. 
If the quantity to be divided is compound, 
then you must range its parts according to the 
dimensions of some one of its letters, as in the 
following example. In the dividend a 2 -j- 2 ab 
-J- b 2 , they are ranged according to the dimen- 
sions of a , the quantity a 2 , where a is of two 
dimensions, being placed first, 2 ab, where it is 
of one dimension, next, and b 2 , where a is not 
at all, being placed last. The divisor must be 
ranged according to the dimensions of the same 
letters ; then you are to divide the first term of 
the dividend by the first term of the divisor, and 
to set down the quotient, which in this example 
is a ; then multiply this quotient by the whole 
divisor, and subtract the product from the divi- 
dend, and the remainder shall give a new divi- 
dend, which in this example is ab -{-■ b 2 , thus, 
a -J- b) a 2 -|- 2 ab -| - b 2 (a b 
a 2 -j- ab 
ab -f b 2 
ab -f b 2 
0 0 
Divide the first term of this new dividend by 
the first term of the divisor, and set down the 
quotient (which in this example is b) with its 
proper sign. Then multiply the whole divisor 
by this part of the quotient, and subtract the 
product from the new dividend ; and if there is 
no remainder, the division is finished. If there 
is a remainder, you are to proceed after the same 
manner till no remainder is left, or till it appears 
that there will be always a remainder. 
Some examples will illustrate this operation, 
EXAMPLE i. 
a 4 . 4 ) a 2 - b 1 la- b 
a \+ ab 
— ab — b 2 
— ab — b 2 
0 0 
EXAMPLE II. 
a — b) aaa — 3 aab -{- 3abb — bbb (aa — 2 ab 4“ bb 
aaa — aab 
2 aab ■ 
— 2 aab 4- 2abb 
3 abb — bbb 
abb 
abb 
bbb 
bbb 
EXAMPLE HI. 
b) aaa — bbb (aa 4™ ab 4~ ^ 
ana — aab 
aab 
aab 
bbb 
abb 
abb — bbb 
abb ■ — bbb 
0 
0 
EXAMPLE IV. 
3a -— 6 ) Gaaaa — 96 ('2aaa -[- 4 aa -j- 8 a -j- It? 
Gaaaa — 12 aaa 
12 aaa — 96 
1 2 aaa - — 24 a a 
24 aa — 96 
24aa — 48a 
48a 
48a 
96 
S 6 
0 O 
It often happens that the operation may be 
continued without end, and then you have an 
infinite series for the quotient ; and. by compar- 
ing the first three or four terms, you may find 
what law the terms observe; by which means, 
without any more division, you may continue 
the quotient as far as you please. Thus, in di- 
viding 1 by 1 — a, you find the quotient to be 
1 4 . a 4- aa 4- aaa 4- a aaa 4 - See. which series 
can be continued as far as you please, by adding 
the powers of a. 
The operation is thus : 
1 — a) 1 (1 a -j- aa 4- aaa, &C. 
1 — a 
4“ a 
4-a 
t 
t 
aaa 
aaa — aaaa 
4 - aaaa, 
Another Example : 
2xx 2a 3 2x 4 
a 4 * x ) &a -a a (a — x —I— — * ; 1 v *“— 1 o*.C* 
1/1 x ' a a a J 
aa 4 ~ ax 
ax 4” xx 
ax — xx 
2aa 
, 2 .v 3 
4 - 2xx 4~ — - 
a 
2a 3 
a 
2 x 3 
4 ~ 
2a 4 
See. 
In this last example the signs are alternately 
-f- and v — , the co-efficient is constantly 2 after 
the first two terms, and the letters are the 
powers of a and a ; so that the quotient may be 
continued as far as you please without any more 
division. " 
But in division, after you come to a remain- 
der of one term, as 2aa in the last example, it 
is commonly set down with the divisor under it, 
after the other terms, and these together give 
