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401 Fundamen-of geometry and algebra, there may be an opposition terms in the quantities to be added may be united, fo Fundamental opera- or contrariety in the quantities, analogous to that of as to render the expresfion of the fum more simple. addition and subtraction ; and the signs to and - may
tions. very conveniently be used to express that contrariety.
Prob. II. To fubtralt Quantities
subtracted (by Prob. I.); the fum arising by this ad.
dition is the remainder. CHA P. I.
79b-165c Secr. 1. Fundamental Operations.'
From 52-76 +9c+8
Subt. 2a-4b +90-d
When a positive quantity is to be fubtracted, the
the negative part of a quantity is to be subtracted, let
and be subtracted from a, the remainder, according Cafe 1. To add terms that are like and have like to the rule, is a-c+d. For if ç is subtracted from ligns.
a, the remainder is a-c (by Def. 3.); but this is too
by d; and d being added, it is a--c+d, according to
Otherwise, If the quantity d be added to these two
quantities a and c-d, the difference will continue the
same; that is, the excess of a above cmd is equal to
cess of a +d above c, which plainly is a +1-4, and is
Prob. III. To multiply Quantities.
prefix the sign of the greater to the remainder, and two terms to be multiplied are like, the lìgn of the
product is + ; but when the signs are unlike, the
sign of the product is --,
Cafe 1. To multiply two terms.
Rule. Find the sign of the product by the general rule; +5bc
after it place the product of the numeral coefficients, Cafe 3. To add terms that are unlike.
and then set down all the letters one after another, as
in one word. Rule. Set them all down, one after another, with their figns and coefficients prefixed.
The reason of this rule is derived from Def. 6. and
from the nature of multiplication, which is a repeated Compound quantities are added together, by uniting addition of one of the quantities to be multiplied as
the several terms of which they consist by the pre- often as there are units in the other. Hence also the
letters in two terms multiplied together may be placed
in any order, and therefore the order of the alphabet Examp. The sum of 3 7xy-ab + 15
is generally preferred. 9cd4xy~401
Cafe 2. To multiply compound quantities. is 4ab-300+15-mn
Rule. Multiply every term of the multiplicand by all
the terms of the multiplier, one after another, accordThe rule for cafe 3. may be considered as the gene- ing to the preceding rule, and then collect all the ral rule for adding all algebraical quantities whatsoever; products into one fum; that sum is the product reand, by the rules in the two preceding cases, the like quired. Vol. I. Part II,
Fundamen- Examp. Mult. 2a+36
ber of terms of a compound quantity, to denote those Fundamen. By Zax-4by
which are understood to be affected by the particular tal opera. tibns.
Thus, in the last example, it shows that the terms
fa and mob, and also cand -d are all affected by the
fign (x). Without the vinculum, the expreffion Prod. baax +gabx-Suby— 12bby
a--bXcod would mean the excess of a above bc and Mult. amb
d; and a-bxced would mean the excess of the proВу co
duct of a b by c, above d. Thus allo, a +b) ex
preffes the second power of a th, of the product of accach
that quantity multiplied by itself; whereas a+b would -adt db
express only the suin of a and b?; and so of others.
By some writers a parenthesis ( ) is ufed as a vincum
lum, and (a+b) is the same thing as a 76).
PROB. IV. To divide Quantities.
+; if they are unlike, the fign of the quotient is
This rule is easily deduced from that given in Prob.
From Def. 8. the quotient of any iwo quantities.
Def. 12. When several quantities are multiplied to- Case I. When the divisor is fimple, and is a factor
13. The products arising from the continual multi- covered by inspection ; for then the co-efficent of the
term of the dividend.
punged out of each term in the dividend, and the
coefficients of each term to be divided by the coindex, or exponent, and from it the power is denomina
efficient of the divisor; the quantity resulting is ted. Thus,
Ex. a) ab(b. 2aab) 6a3bc--4aobdm (326-2dın.
The reason of this is evident from the nature of di-
vifion, and from Def. 6. Note. It is obvious from
Case II. When the divisor is simple, but not a
ing to Def. 8. viz. by placing the dividend above a
line and the divisor below it.
Thus, the quotient of zab' divided by 2mbc is the
Such expreflions of quotients may often be reduced
to a more fimple form, as shall be explained in the sea
cond part of this chapter.