ALGEBRA 
283 
ab fceA~!>dzzz 56 -f-6—3 5—2 7 • 
«~M , 3^—2C 8-4-7 . ai— _ i 5_ l 9_ , 
aJ-d^ 6—1 8—5 — ~ 5 ' 3*”" 
d 2 X a—c—3ce 2 +^ 3 =:25 X a—iS-J-x 25=: 50— 18-4-125 
= 157- 
Axioms. 
If equal quantities be added to equal quantities the 
wholes will be equal. 
If equal quantities be taken from equal quantities the 
remainders will be equal. 
If equal quantities be multiplied by the fame or equal 
quantities the products will be equal. 
If equal quantities be divided by the lame or equal quan¬ 
tities the quotients will be equal. 
If the fame quantity be added to and fubtracted from 
another the value of the latter will not be altered. 
If a quantity be both multiplied and divided by another 
its value will not be altered. 
ADDITION of ALGEBRAICAL QUANTITIES. 
To add quantities together is to conned! them with their 
proper figns, uniting into one fum thofe that can be fo 
united. £x. 1. If the following quantities are to be added: 
Ex. 2. 
a fib — c 
d-V+f 
ax 
— by 
-K 
— cd 
Sum afzb — cf-d—$eff 
Sum is ax — byfp— cd 
It is immaterial in what order the quantities are fet 
down, if we take care to prefix to each its proper fign. 
When any terms are fimilar they may be incorporated, and 
the general expreflion for the fum fhortened. 
Firft, When fimilar quantities have the fame fign, their 
fum is found by taking the fum of the coefficients with 
that fign and annexing the common letters. 
Ex. 3. 
S a —i h 
Sum pa 
o b 
Ex. 4. 
Sum 
4 ale—lobde 
6 ale —■ 9 bde 
11 adc — 3 bde 
2 1 a l c —2 zbde 
The reafon is evident; 5a to be added together with 4 a 
to be added makes 9 a to be added ; and f to be fubtradl- 
ed together with f to be fubtrafted is 10M0 be l'ubtrafted. 
. Secondly, If fimilar quantities have different figns, their 
fum is found by taking the difference of the coefficients 
with the fign of the greater and annexing the common let¬ 
ters as before. 
Ex. 5. 7«+3 5 
Sum za — 6b 
In the firft part of the operation we have 7 times a to~ 
add, and 5 times a to take away, therefore upon the whole 
we have za to add. In the latter part, we have 3 times b 
to add, and 9 times b to take away, i. e. we have upon the 
whole 6 times 1 to take away; and thus the fum of all the 
quantities is za — 6b. 
Ex. 6. afb 
a—b 
Sum 2 a * 
If feveral fimilar quantities are to be added together, fome 
with pofitive and fome with negative figns, take the diffe¬ 
rence between the fum of the pofitive Coefficients and the 
fum of the negative, prefix the fign of the greater fum 
and annex the common letters. 
Ex. 7. 
— $a" f 6 bcf ze ~—15 
—4a 2 —9 be — 1 Of 2 4 - 21 
Sum —6a 2 -}- be — 9 e 2 +i6 
The method of reafoning in this cafe is the fame as in the 
laft example. 
SUBTRACTION. 
Subtraction, or the taking away of one quantity from 
another, is performed by changing the fign of the quanti¬ 
ty to be fubtracted, and then addingit to the other. 
Ex. 1. Prom 2 bx take cy, and the difference is properly 
reprefented by 2 bx — cy ; becaufe the fign — prefixed to 
cy, fhews that it is to be fubtrafted from the other; and 
2 bx—cy is the fum of 2 bx and — cy. 
Ex. 2. Again from 2bx take — cy, and the difference is 
2 bxfcy, becaufe tbx^zzbxfcy — cy. Take away —cy 
from thefe equal quantities, and the differences will be equal, 
i. e. the difference between 2 bx and — cy=zzbxfcy, the 
quantity which arifes from adding fey to 2 bx. 
Ex. 
F rom 
Take 
Difference 
afb 
a—b 
* fib 
Ex. 
From 
Take 
Diff. 
6a —12 b 
■e,a —10 b 
Ta—Tfl 
Ex. 5. 
From ^arfiyab —6x7 
Take ixalf6ab —4x7 
Ex. 6. 
4<z— Z,bf6c —11 
lox-f- a —15—2v 
Diff. —6 a 2 —2 ab —227 
—iox-fatz—3^-}-4-f6c-}-27 
MULTIPLICATION. 
The multiplication of fimple algebraical quantities is 
performed thus: ay.b or ab reprefents the product of a 
multiplied by b ; abc the product of the three quantities a , 
b, and c. It is alfo indifferent in what order they are pla¬ 
ced, aXb and bXa being equal. For 1 X a~a X 1 or one 
taken a times is the fame with a taken once; alfo b taken 
a times, or bx a, is b times as great as 1 taken a times, 
and a taken b times, or a X b, is b times as great as a taken 
once; therefore bx a—a X b. Alfo abc=cabz=zbca~acb. 
Sec. for, as in the former cafe, »X«Xt*X^X 1 and 
cX a Xb E c times as great as iX^Xb, alfo aXbXe is 
c times as great as a X b X 1 , therefore aXbx c~cxaxb. 
To determine th e fign of the product obferve the fol¬ 
lowing rule: — If tire multiplier and multiplicand have the 
fame fign, the product is pofitive: if they have different 
figns, it is negative. 
Firft, -f-aX-H—-fbecaufe in this cafe a is to be 
taken pofitiveiy b times, therefore the product ab muff be 
pofitive. 
Secondly, — axfb~ — ab-, becaufe —a is to be taken 
b times, that is, we mult take — ab. 
Thirdly, ~}-<2X— b— —f° r a quantity is faid to be 
multiplied by a negative number — b, if it be fubtracted 
b times; and a fubtraCted b times is — ab. 
Fourthly, — ax — b—fab. Here — a is to be fub¬ 
tracted b times, that is —ab is to be fubtraCted, but fub- 
traCtihg —ab is the fame as adding fab-, therefore we 
have to add fab. 
The fecond and fourth cafes may be thus proved: a —. 
<2=0; multiply both fides by b, and axb together with — a 
Xb mud be equal to bx o, or nothing; therefore, —a mul¬ 
tiplied by b muff give — ab, a quantity which when added 
to ab makes the fum nothing. Again, a — a— o; multiply 
both fides by — b, then —ab together with — ax—b muft 
be =0, therefore.— aX — b—fab. 
If the quantities to be multiplied have coefficients, thefe 
muft be multiplied together as in common arithmetic; 
the fign and the literal produCt being determined by the 
preceding rules. Thus 30X5^— t^ab-, becaufe 3 X<2X 
5 X fc=3 X 5 X X b—i:$ab ; 4.x X —117=—44x7 ; —o£x 
— ^c—f^fc ; •— 6dxfm~ —24 md. 
i he powers of the fame quantity are multiplied toge 
ther by adding the indices; thus a^x^—eP, for aaxaaa 
—aaaaa. In the fame manner a m Xa n —a m * n ; —3«Vx 
s axy*= — i^xf. 
If the multiplier or multiplicand conlift of feveral 
terms, each term of the latter muft be multiplied by eve¬ 
ry term of the former, and the fum of all the products 
taken, for the whole product of the two quantities. 
Mult, 
