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III. The fign

(minus) denotes, that the quantity, before which it is placed, is to be fubtracted. Thus, a-b denotes the excefs of a above b; 6-2 is the excess of 6 above 2, or 4. Note, Thefe characters,

and, from

their extensive use in algebra, are called the figns; and the one is faid to be oppofite or contrary to the other. IV. Quantities which have the fign + prefixed to them, are called pofitive or af firmative; and fuch as have the fign prefixed to them are called negative. V. Quantities which have the fame fign,

either or, are alfo faid to have like +

figns,and thofe which have different figns,

are faid to have unlike figns. Thus +a, +b, have like figns, and a, ―ç, are faid to have unlike figns. VI. The juxtaposition of letters, as in the fame word, expreffes the product of the quantities denoted by these letters. Thus, ab expreffes the product of a and b; bcd expreffes the continued product of b, c,

and

and d. The fign × also expreffes the product of any two quantities between which it is placed.

VII. A number prefixed to a letter is called a numeral coefficient, and expreffes the product of the quantity by that number, or how often the quantity denoted by the letter is to be taken. When no number is prefixed, unit is understood. VIII. The quotient of two quantities is denoted, by placing the dividend above a fmall line, and the divifor below it. Thus,

19

3

is the quotient of 18 divided by 3, or 6;

is the quotient of a divided by b. This

expreffion of a quotient is also called a fraction.

IX. A quantity is faid to be fimple which confifts of one part or term, as +a, -abc; and a quantity is laid to be compound, when it confifts of more than one term, connected by the figns + or -. Thus, a+b, a-b+c, are compound quantities. If there are two terms, it is

called

called a binomial; if three, a trinomial, &c.

X. Simple quantities, or the terms of compound quantities, are faid to be like, which confift of the fame letter or letters, equally repeated. Thus, +ab, 5ab, are like quantities; but +ab, and +aab, are unlike.

XI. The quality of two quantities is expreffed, by placing the sign = between them. Thus, x+a=b-c, means that the sum of x and a is equal to the excefs of b above c.

When quantities are confidered abstractly, the + and denote addition and fubtraction only, according to Def. II. III. and the terms positive and negative exprefs the fame ideas. In that cafe, a negative quantity by itself is unintelligible. The fignalfo is unneceffary before fimple quantities, or before the leading term of a compound quantity which is not negative; though, when fuch a quantity or term is to be added to another,+ must be placed before

fore it, to express that addition; and hence in Def. II. it is faid, that is understood, when no fign is expreffed.

In geometry, however, and in certain applications of geometry and algebra, there may be an oppofition or contrariety in the quantities, analogous to that of addition and subtraction; and the figns + and may very conveniently be used to exprefs that contrariety. In fuch cafes, negative quantities are understood to exist by themfelves; and the fame rules take place in operations into which they enter, as are used with regard to the negative terms of abstract quantities.

When, therefore, in the following elementary rules, negative quantities are introduced as examples, they are to be underfood, either as fubtracted from pofitive quantities not expreffed; or, as having fuch an oppofition as has now been mentioned, to other quantities marked with the fign.'

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A

of Axioms.

LGEBRAICAL reasonings are founded on a set of first principles, not lefs evident than the axioms of Geometry. They are, indeed, fo fimple, that it is unneceffary, in this place, either to enumerate, or to illuftrate them. Befides, a general axiom is not more evident than any particular propofition comprehended in it.

CHAP.

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