A L G £ B it A, 
f-9 
Algetard, as ha* been already observed, is 
called an Universal Arithmetic, and it pro- 
ceeds by operations and rules similar to those 
in common arithmetic, founded -upon the same 
principles. This, however, is no argument 
I against its usefulness or evidence ; since arith- 
metic is not to be the -less Valued that it is 
common, and is allowed to be one of the 
most clear and evident of the sciences. But 
as a number of symbols are admitted into this 
science, being necessary for giving it that ex- 
tent and generality which is its greatest excel- 
lence ; the import of those symbols is to be 
clearly stated, that no obscurity or error may 
arise from the frequent use and complication of 
them. Thus, 
The relation of equality is expressed by the 
sign — ; thus to express that the quantity repre- 
sented by a is equal to that 'which is represented 
by l>, we write a ~ b. But if we would express 
that a is greater than b, we write a ;> b ; and if 
we would express algebraically that a is less 
than b, we write a <j b. 
Quantity is what is made, up of parts, or is 
capable of being greater or less. It is increased 
by Addition, and diminished by Subtraction ; 
which are therefore the two primary operations 
that relate to quantity. Hence it is, that any 
quantity may be supposed to enter into alge- 
braic computations two different ways which 
have contrary effects; either as an increment, 
or as a decrement ; that is, as a quantity to be 
added, or as a quantity to be subtracted. The 
sign -j- (plus) is the mark of Addition, and the 
sign — (minus) of Subtraction. Thus the quan- 
tity being represented by a, or -j- ^ imports that 
a is to be added, or represents an increment ; 
I but — a imports that a is to be subtracted and 
represents a decrement. When several such 
quantities are joined, the signs serve to shew 
! which arc to be added, and which are to be sub- 
i traded. Thus -j- a -(- b denotes the quantity 
that arises when a and b are both considered as 
| increments, and therefore expresses the sum of 
a and b. But -j- a — b denotes the quantity that 
arises when from the quantity a the quantity b 
1 is subtracted ; and expresses the excess of a 
I above b. When a is greater than b, then a — b 
is itself an increment ; when a =t b, then a — L 
— 0 ; and when a is less than b, then a — b is 
itself a decrement. 
As Addition and Subtraction are opposite, or 
1 an increment is opposite to a decrement, there 
! is an analogous opposition between the affec- 
tions of quantities that are considered in the 
mathematical sciences. As between excess and 
defect ; between the value of effects or money 
; due to a man, and money due by him ; a line 
drawn towards the right and a line drawn to 
I the left ; gravity and levity ; elevation above the 
i horizon and depression below it. When two. 
quantities equal in respect of magnitude, but of 
those opposite kinds, are joined together, and 
j conceived to take place in the same subject, they 
j destroy each other’s effect, and their amount is 
j nothing. Thus 100/. due to a man, and 100/. 
due by him, balance each other ; and in esti- 
| mating his stock may be both neglected. 
A quantity that is to be added is likewise 
called a positive quantity ; and a quantity to 
be subtracted is said to be negative : they are 
equally real, but opposite to each other, so as 
to take away each other’s effect, in any opera- 
! {ion, when they are equal as to quantity. Thus 
I 3 — 3=0, and a — a = 0. But though -j- a 
and — a are equal as to quantity, we do not 
1 suppose in Algebra that a — — a ; because 
; to infer equality in this science, they must not 
only be equal as to quantity, but of the same 
quality, that in every operation thd one may 
have the same effect as the other. A decrement 
may be equal to an increment ; but it has in all 
operations a contrary effect. It is on account 
•Von. I. 
of this contrariety that a negative quantity is 
said to be less than nothing, because it is oppo- 
site to the positive, and diminishes it when 
joined to it, whereas the addition of 0 has no 
effect. But a negative is to be considered no 
less as a real quantity than the positive. Quan- 
tities that have no sign prefixed to them are 
understood to be positive. 
The number prefixed to a letter is Called the 
numeral co-efficient, and shews how often the 
quantity represented by the letter is to be taken. 
Thus 2a imports that the quantity represented 
by a is to be taken twice ; 3 a that it is to be 
taken thrice, and so on. When no number is 
prefixed, unit is understood to he the co-efficient. 
Thus 1 is the co-efficient of a or of b. 
Quantities are saief to be like or similar, that 
are represented by the same letter, or letters 
equally repeated. Thus, -}- 3 a and — 5a are 
like ; but a and b, or a and aa, are unlike. 
A quantity is said to consist of as many terms 
as there are parts joined by the signs -j- or — ; 
thus a — b consists of two terms, and is called 
a binomial ; a -j- b -j- c consists of three terms, 
and is called a trinomial. These are called 
compound quantities : a simple quantity con- 
sists of one term only, as -j- a , or 4~ ab, or 
-j- abc. 
Of Addition. 
Case I. To add quantities that are like and have 
like signs. 
Rule. Add together the co-efficients, to their 
sum prefix the common sign, and subjoin the 
Common letter or tetters. 
EXAMPLES. 
To -j- 5a to — Gb to a -j- b 
Add -}- 4 a add — 2 b add 3 a -)- 5b 
Suffi -j- 9a Sum — S b Sum 4a -j- Gb 
To 3 a — 4 k 
add 5a — 8k 
Sum 8 a — 12k 
Case It. To add quantities that are like but have 
unlike signs. 
Rule. Subtract the lesser co-efficient from the 
greater, prefix the sign of the greater to the 
remainder, and subjoin the common letter or 
letters. 
EXAMPLES. 
To — 4 a 
Add -j- 7 a 
4" 5b — Gc 
— 3b -f- 8c 
Sum 4 - 2 b 4~ 2c 
To a 6x — 5y -[- 8 I 2 a — 2b 
5a — 4x -{- 4 y — 3 j — 2 a -j- 2b 
Add 
Sum 
4 a -j- 2x — y -J- 5 
0 . 
O 
This rule is easily deduced from the nature of 
positive and negative quantities. 
If there are more than two quantities to be 
added together, first add the positive together 
into one sum, and then the' negative (by Case I). 
Then add these two stuns together (by Case II). 
8 abx 
— 7 abx 
-j- 1 0 abx 
. • — 1 2 abx 
Sum of the positive -j- IS^k 
S um of the negative — 19 abx. 
Sum of all — abx 
Case III. To add quantities that are unlike. 
Rule. Set them all down one after another, 
with their signs and Co-efficients prefixed. 
G 
s if Atoms. 
To 4- 2 a ft* 
Add -j- 3 b — 4 k 
Sum 2a -\~ 3b 3 a — 4k 
To 4 a -j- 4b -j- 3c 
Add — 4k — 4y -J- 3z 
Sum 4a -}- 4b -j- 3c — 4.v — 4y -{- 3s 
Of Subtraction. 
General Rule. Change the signs of the quan- 
tity to he subtracted into their contrary signs, 
and then add it so changed to the quantity 
from which it was to be subtracted (by the 
preceding rules) : the sum arising by this ad- 
dition is the remainder. For, to subtract any 
quantity, either positive or negative, is the 
same as to add the opposite kind. 
EXAMPLES. 
From 
Subtract 
-}- 5a 
-f 3a 
8 a — 
3a -f- 
7 b 
4b 
Remainder 5a — 3<i, or 2 a 5a — 116 
From 2a — 3k 4* 5y — G 
Subtract -)- 4 k - j- 5y 4“ -4 
Remainder 4 a — 7k 0 — 10 
It is evident, that to subtract or take away A 
decrement is the same as adding an equal in- 
crement. If we take away — b from a — 
there remains a ; and if we add -|- b to a — > bf 
the sum is likewise a. In general, the subtrac- 
tion of a negative quantity is equivalent to add- 
ing its positive value. 
Of Multiplications 
In Multiplication the general Rule for the 
signs is. That when the signs of the factors' are 
like (i. e. both -j-> of both — ) the sign of the 
product is 4 ; hut when the signs of the factors 
are unlike, the sign of the product is — . 
Case I. When any positive quantity, -J- a , is 
multiplied by any positive number, 4- «, the 
meaning is, that -j- a is to be taken as many 
times as there are units in n ; and the product is 
evidently na. 
Case If. When — a is multiplied by n, then 
— a is to be taken as often as there are units irt 
n, and the product must be — na. 
Case III. Multiplication by a positive numbei' 
implies a repeated addition : but multiplication 
by a negative implies a repeated subtraction. 
And when 4- a is to he multiplied by — n, the 
meaning is, that 4 ( a is to be subtracted as often- 
as there are units in n. Therefore the product 
is negative, being — na. 
Case IV. When — a is to be multiplied by 
— n, then — a is to be subtracted as often as. 
there are units in »; but to subtract — a is equi- 
valent to adding -f- «■> consequently the product 
is -j- na. 
The lid and IVth Cases may be illustrated in 
the following manner. 
By the definitions, a — a ~ 0 ; therefore,, 
if we multiply -j- a — a by n, the product must 
vanish or he 0, because the factor a — a is 0_ 
The first term of the product is-f-na (by Case I.) 
Therefore the second term of the product must 
be — na, which destroys 4- m; so that the. 
whole product must he -j- na — na =0. There- 
fore — a multiplied by 4- » gives — na. 
In like manner, if we multiply 4 -a — a by 
— n, the first term of the product being — net , 
the latter term of the product must be -{- na* 
because the two together must destroy eacii 
other, or their amount be 0, since one of the 
factors (viz. a — , a) is 0. Therefore — a multi- 
plied by — n must grfe -j- na. 
In this genera) doctrine the multiplicator 
always considered as a number. A quantity of 
any kind may be multiplied by a number ; bttt/ 
