3?f 
ALGEBRA. 
Mult, a fb 
By cfd 
Prod. acfbcfatifbd 
Here afb is to be added to itfelf cfd times, i. e. c times 
and d times. 
Mult. 
By c—d 
Prod. acfbc — ad—bd 
Here afb is to be taken c—d times, that is c times want¬ 
ing' d times, or c times poiitively and d times negatively. 
Ex. i. Mult. 
By 
afb 
afb 
a" fab 
-f -abfb* 
Prod. afzabfV 
Muir." 
By 
±tx. 2. Mult. afb 
By a—b 
ad fab 
— ab- 
Prod. 
Ex. 
3 dr—sbd 
bd 
-j-i 2 adbd —20 b 2 d 2 
Prod. — L^a^f^adbcl —20 Ird* 
Scholium. The method of determining the fign of a 
prodtiit from the confideration of abftradf quantities has 
been found fault with by fome algebraical writers, who 
contend that — a, without reference to other quantities, 
is Imaginary and confequently not the objefl: of reafon or 
demonftration. In anfwer to this objeftion we may ob- 
ferve ; that whenever we make ufe of the notation — a, 
and fay it fignifies a quantity to be fubtradted, we make 
a tacit reference to other quantities, and mean, that where - 
ever a is concerned it is to be taken from the collective 
furn of thofe magnitudes with which it is connected; and 
when we fpeak ot —a being taken b times, or a fubtraCt- 
ed b times, we mull underhand that it is to be taken b times 
from fome other quantities. —If in the folution of any pro¬ 
blem it fhould appear that a quantity refults which is to be 
i'ubtracted, and there is nothing to fubtraft it from, we 
mu ft either conclude-that the problem itfelf involves an 
abfurdity, or, that fome fuppofition made in the opera¬ 
tion is inconlihent with the conditions of the problem. 
Vid. Simplon’s Alg. p-23- 
DIVISION. 
To divide one quantity by another, is to determine how 
often the latter is contained in.the former, or what quan¬ 
tity multiplied by the latter will give the former. Thus 
to divide ab by a is to determine how often a mud be taken 
to make up ab, that is, what quantity multiplied by a will 
oive ab , which we know is b. From this confideration 
are derived all the rules for the divifion of algebraical 
quantities. 
If the divifor and dividend be affected with like figns, 
the fignof the quotient is +; but, if their figns be unlike, 
the fign of the quotient is —. If —ab be divided by — a, 
the quotient is -f b, becaufe — ay.fb gives— ab-, and a 
fimilar proof may be applied to the other cales. 
In the divifion of fintple quantities, if the coefficient and 
literal product of the divifor be found in the dividend, the 
other part of the dividend with the fign determined by die 
lafl rule is the quotient. 
Thus ^t c r — r, becaufe ab multiplied by c gives abc. 
ab 
If we firft divide by a and then by b, the refult will be the 
fame: for ~=.bc, and-^=c as before. 
a b 
Cor. Hence any power of a quantity is divided by any 
either power of the fame quantity by taking the index of 
the divifor from the index of the dividend. 
a 5 „ a s 1 _ , a m 
Thus, — ~a 2 ; —— — =« 3 ; — — a m -”- 
a 3 a s a 3 a a 
If only a part of the product which forms the divifor be 
contained in the dividend, the divifion muft be reprefent- 
ed as follows, and the quantities contained in both the di¬ 
vifor and dividend expunged. 
• Thus, i^a 3 b 2 c divided by—3 a 2 bx or— 
— la^bx x - 
Firft, divide by—3^ and the quotient is— sabc-, this 
quantity is ftill to be divided by x-, and, as * is not contain¬ 
ed in it, the divifion can only be reprefented in the ufual 
way, that is, ——— is the quotient. 
x 
If the dividend confift of feveral terms, and the divifor 
be a limple quantity, every term of the dividend mud be 
divided by it. 
a 3 x 2 —c abx 3 —6 ax A 
Thus, -——- —a — fx —6 x'. 
ax 
When the divifor alfo confifts of feveral terms, arrange 
both the divifor and dividend according to the powers of 
fome one letter contained in them, then find how often the 
firft term of the divifor is contained in the firft term of the 
dividend, and write down this quantity for the firft term 
in the quotient; multiply the whole divifor by it, fubtraCl 
the product from the dividend, and bring down to the re¬ 
mainder as many other terms of the dividend as the cafe 
may require, and repeat the operation till ail the terms are 
brought down. 
Ex. If a* — zabfb* be divided by a — b, the operation 
will be as follows: 
a — b)ad — iabfb' 1 '(a~~b 
a“—ab 
—abfb' 1 
— abfb* 
The reafon of this, and the preceding rule, is, that as 
the whole dividend is made up of all its parts, the divifor 
is contained in the whole as often as it is contained in ail 
the parts. In the preceding operation we inquire firft how 
often a is contained in a 1 , which gives a for the firft tepm 
of the quotient; then multiplying the whole divifor by it, 
we have a*—ab to be fubtrafted from the dividend, and the 
remainder is .— abfb\ with which we are to proceed as 
before. 
On the Transformation of Frablions to others of equalValue. 
If the figns of all the terms both in the numerator and 
denominator of a fraction be changed, its value will not be 
-ab -f -ab 
■fa 
altered. For- 
If the numerator and denominator of a fraction be both 
multiplied or both divided by the fame quantity, its va¬ 
lue is not altered. For . Hence a fraction is redu¬ 
ce b 
ced to its-lowed terms, by dividing both the numerator 
and denominator by the greateft quantity that meafures 
them both. The greateft common, meafureof two quanti¬ 
ties is found by arranging them according to the powers 
of fome letter, and then dividing the greater by the lets, 
and the preceding divifor always by thelaft remainder, till 
the remainder is nothing; the laft divifor will be the great¬ 
eft common meafure required. 
Let a and b be the two quantities, and let b b)a(p 
be contained in a, p times with a remainder c; c)b(q 
again let c be contained in b, q times with a re- 
mainder d, and fo on till nothing remains ; let . 
d be the laft divifor, and it will be the greateft o 
common meafure of a and b. The truth of this rule de¬ 
pends upon thefe two principles ; 
1. If one quantity meafure another, it will alfo meafure 
any multiple of it. Let x meafure y by the units in n, then 
it will meafure cy by the units in nr.. 
2. If a quantity meafure two others, it will meafure their 
funi or difference. Let a be contained in x, m times, and 
in 
