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Part I.

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
In such cases, negative quantities are understood to General Rule. Change the signs of the quantity to be
exist by themselves; and the fame rules take place in fubtracted into the contrary ligns, and then add it,
operations into which they enter, as are used with regard so changed, to the quantity from which it was to be
to the negative terms of abstract quantities.,

subtracted (by Prob. I.); the fum arising by this ad.

dition is the remainder. CHA P. I.

Examp. From


79b-165c Secr. 1. Fundamental Operations.'

The fundamental operations in algebra are the same

+ 2a

as in common arithmetic, Addition, Subiraltion, Mul-

From 52-76 +9c+8
tiplication, and Division; and from the various combi.

Subt. 2a-4b +90-d
nations of these four, all the others are derived.
PROB. I. To add Quantities.

Rem. 32~36o*+8+1
Simple quantities, or the terms of compound quantities,

When a positive quantity is to be fubtracted, the
to be added together, may be like with like fignis, like rule is obvious from Def. 3. In order to show it, when
with unlike hgns, or they may be unlike.

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
Rule. Add together the coefficients, to their fum pre. {mall, because c is fubtracted instead of c-d, which is

fix the common fign, and subjoin the common letter less than it by d; the remainder therefore is too small
or letters.

by d; and d being added, it is a--c+d, according to

the rule.
Examp. To 5ab 3aa--ab

Otherwise, If the quantity d be added to these two
- 2ab

quantities a and c-d, the difference will continue the

same; that is, the excess of a above cmd is equal to
the excess of a td above c-d+d, that is, to the ex-

cess of a +d above c, which plainly is a +1-4, and is
Cafe 2. To add terms that are like, but have unlike therefore the remainder required.

Prob. III. To multiply Quantities.
Rule. Subtract the less coefficient from the greater ; General Rule for the Signs. When the figns of the

prefix the sign of the greater to the remainder, and two terms to be multiplied are like, the lìgn of the
subjoin the common letter or letters.

product is + ; but when the signs are unlike, the
Examp. 4a
+7bc --5ab

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.

Mult. ta




Examp. 2a + 36

- igbc


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
ceding rules.

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,

3 E





+ bc

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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.

fgn connected with it.
baax +9abx
mmt mux

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
Prod. ac-cb-adt db

lum, and (a+b) is the same thing as a 76).
of the general Rale for the signs,

PROB. IV. To divide Quantities.
The reason of that rule will appear by proving it, as General Rule for the Signs. If the figns of the divisor
applied to the laft mentioned example of a--b multiplied and dividend are like, the sign of the quotient is
by cd, in which every case of it oceurs.

+; if they are unlike, the fign of the quotient is
Since multiplication is a repeated addition of the
multiplicand as often as there are units in the multi-
plier, hence, if amb is to be multiplied by c, a. 111.; for, from the nature of divifion, the quotient

This rule is easily deduced from that given in Prob.
must be added to itself as often as there are units in muit' be such a quantity as, multiplied by the divisor,
c, and the produ& therefore must be ca~-cb (Prob. 1.). fhall produce the dividend with its proper lign.
But this product is too great ; for amb is to be

From Def. 8. the quotient of any iwo quantities.
multiplied, not by c, but by cod only, which is the
excess of c above 1; 3 times ab therefore, or da-db, may be expressed, by placing the dividend above a line
excess of c above d; d times a-6 therefore, or dadb, and the divisor below it. But a quotient may often
has been taken too much; hence this quantity must be expressed in a more fimple and convenient form, as.
be subtracted from the former part of the product, and will appear from the following distinction of the
the remainder, which (by Prob. II.) is cacb-da+db,

will be the true product required.

Def. 12. When several quantities are multiplied to- Case I. When the divisor is fimple, and is a factor
gether, any of them is called a factor of the produ&. of all the terms of the dividend. This is easily dis-

13. The products arising from the continual multi- covered by inspection ; for then the co-efficent of the
plication of the fame quantity are called the powers of divisor measures that of all the terms of the dividend,
that quantity, which is the root. Thus, aa, aaa, aaaa, and all the letters of the divisor are found in every
&c. are powers of the root a.

term of the dividend.
14. These powers are expressed, by placing above Rule. The letter or letters in the divisor are to be ex.
the root, to the right hand, a figure, denoting how
often the root is repeated. This figure is called an

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,

the quotient.
ift Power of the root a' or a

Ex. a) ab(b. 2aab) 6a3bc--4aobdm (326-2dın.
a, and is other a
wise expressed

The reason of this is evident from the nature of di-

vifion, and from Def. 6. Note. It is obvious from
The ad and 3d powers are generally called the corollary to Prob. III. that powers of the same root
square and cube; and the 4th, sth, and 6th, are alle are divided by subracting their exponents.
Tometimes respectively called the biquadrate, furfolid, Thus a')a} (a a:)a' (a“. Also a*b)a366 (abs.
and cubocube,

Case II. When the divisor is simple, but not a
Cor. Powers of the same root are multiplied by factor of the dividend.
adding their exponents. Thus, a*Xa=a', or euax Rule. The quotient is expressed by a fraction, accord.
80=aaaaa, b3Xb=64.

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
Sometimes it is convenient to express the multipli-

cation of quantities, by setting them down with the fraction

2 mbc.
fign (x) between them, without performing the ope-
ration according to the preceding rules ; thus a*Xb is

Such expreflions of quotients may often be reduced
written instead of a*b; and a-bXced expresses the

to a more fimple form, as shall be explained in the sea
product of a-b, multiplied by cậd.

cond part of this chapter.
Def. 15. A vinculum is a line drawn over any aum- Cafe III. When the divisor is compound.





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