ill. The fign (minus) denotes, that the quantity, before which it is placed, is to be subtracted. Thus, amb denotes the excess of a above b; 6-2 is the excess of 6 above 2, or 4. Note, These characters, + and , from their extensive use in algebra, are called the signs; and the one is said to be oppofite or contrary to the other. IV. Quantities which have the sign + pre fixed to them, are called positive or affirmative ; and such as have the sign prefixed to them are called negative. V. Quantities which have the same fign, either + or -, are also faid to have like Tigns, and thofe which have different signs, are said to have unlike signs. Thus ta +6, have like figns, and ta, -, are faid to have unlike figns. VI. The juxtaposition of letters, as in the fame word, expresses the product of the quantities denoted by these letters. Thus, ab expresses the product of a and b; bcd expresses the continued product of b, c, and 1 and d. The sign X also expresses the produd of any two quantities between which it is placed. VII. A number prefixed to a letter is called a numeral coefficient, and expresses the product of the quantity by that number, or how often the quantity denoted by the letter is to be taken. When no num ber is pretixed, unit is understood. VIII. The quotient of two quantities is de noted, by placing the dividend above a small line, and the divifor below it. Thus, -is the quotient of 18 divided by 3, or 6; 19 3 a is the quotient of a divided by b. This expression of a quotient is also called a fraction. IX. A quantity is said to be simple which consists of one part or term, as ta, -abc; and a quantity is said to be compound, when it consists of more than one term, connected by the signs + or -, Thus, atb, a-b+c, are compound quantities. If there are two terms, it is salled a binomial; if three, a trinomial, called fore &c, X. Simple quantities, or the terms of com pound quantities, are said to be like, taab, are unlike. XI. The quality of two quantities is ex pressed, by placing the sign = between them. Thus, xta=b-c, means that the sum of x and a is equal to the excess of b above c. When quantities are considered abstractly, the + and - denote addition and subtraction only, according to Def. II. III. ; and the terms positive and negative express the same ideas. In that case, a negative quantity by itself is unintelligible. The sign + also is unnecessary before simple quantities, or before the leading term of a compound quantity which is not negative; though, when such a quantity or term is to be added to another, t must be placed be (9) fore it, to express that addition ; and hence in Def. II. it is said, that + is understood, when no sign is expressed. In geometry, however, and in certain applications of geometry and algebra, there may be an opposition or contrariety in the quantities, analogous to that of addition and subtraction; and the signs + and may very conveniently be used to express that contrariety. In such cases, negative quantities are understood to exist by themselves ; and the same 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 understood, either as subtracted from positive quantities not expressed ; or, as having such an opposition as has now been mentioned, to other quantities marked with the sign. Of Axioms. A LGEBRAICAL reasonings are foun ded on a set of first principles, not less evident than the axioms of Geometry. They are, indeed, so simple, that it is unnecessary, in this place, either to enumerate, or to illustrate them. Besides, a general axiom is not more evident than any particular proposition comprehended in it. CHAP |