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'HE fundamental operations in Alge

bra are the same as in common Arithmetic, Addition, SUBTRACTION, MULTIPLICATION, and Division; and from the various combinations of these four, all the others are derived.


To add Quantities.

Simple quantities, or the terms of com

pound quantities, to be added together, may be, like with like signs, like with unlike signs, or they may be unlike.

CASE I. To add Terms that are like, and have like signs.

Rule. Add together the coefficients, to their

sum prefix the common fign, and subjuin the common letter or letters.


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CASE II. To add Terms that are like, but have unlike signs. Rule. Subtra&t the less coefficient from the

greater ; prefix the sign of the greater to the remainder, and subjoin the common letter or letters.




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+5bc CASE III. To add Terms that are unlike.

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Rule. Set them all down, one after another,

with their signs and coefficients prefixed.

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Compound quantities are added together,

by uniting the several Terms of which they confift, by the preceding rules.


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The rule for Cafe III.


be considered as the general rule for adding all Algebraical Quantities whatsoever, and, by the rules in the two preceding cases, the like Terms in the quantities to be added may be united, so as to render the expression of the fum more simple.

PROB. II. To subtract Quantities. General Rule. Change the figns of the

quantity to be subtracted into the contrary figns, and then add it, so changed, to the quantity from which it was to be subtracted; (by Prob. I.) the sum arising by this addition, is the remainder.

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When a positive quantity is to be subfracted, the rule is obvious from Def. 3. : In order to fhew it, when the negative part of a quantity is to be subtracted, let c-d be subtracted from a, the remainder, according to the rule, is a-c+d. For, if c is fubtracted from a, the remainder is a-, (by Def. 3.); but this is too finall, because c is subtracted instead of c-d, which is less than it by d; the remainder, therefore, is too small by d; and d being added, it is ac+d, according to the rule.

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 c-d, is equal to the excess , of atd above cd+d; that is, to the excess of etd above c, which plainly is


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atd>, and is therefore the remainder required.

PROB. III. To multiply Quantities. General Rule for the Signs. When the Signs of the two terms to be multiplied arç like, the sign of the product is +; but, when the signs are unlike, the sign of the product is

CASE I. To multiply two Terms. Rule. Find the sign of the produet by the

general rule ; after it place, the product of the numeral coeficients, and then set down all the letters one after another, as in one word.

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Mult. ta
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+35aabx The reason of this rule is derived from Def. 6. and from the nature of multiplication, which is a repeated addition of one of the quantities to be multiplied as often as there are units in the other. Hence also.


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