the letters in two terms multiplied together

may

be placed in

any order, and therefore the order of the alphabet is generally preferred.

CASE II. To multiply compound quantities.

Rule. Multiply every term of the multiplicand by all the terms of the multiplier, one after another, according to the preceding rule, and then collect all the products into one fum; that fum is the product required.

Of the General Rule for the Signs.

The reafon of that rule will appear, by proving it, as applied to the laft mentioned example of a-b, multiplied by c-d, in which every cafe of it occurs.

Since multiplication is a repeated addition of the multiplicand, as often as there are units in the multiplier; hence, if a-b is to be multiplied by c, a-b must be added to itself, as often as there are units in c, and the product therefore must be cả—cb, (Prob. I.).

But this product is too great; for a—b is to be multiplied, not by c, but by c-d only, which is the excefs of c above d; d times a―b, therefore, or da-db, has been taken too much; hence, this quantity must be fubtracted from the former part of the product, and the remainder, which (by Prob. II.) is ca-cb-da+db, will be the true product required.

Def. XII. When feveral Quantities are multiplied together, any of them is called a factor of the product.

XIII. The products arifing from the continual multiplication of the fame quan tity, are called the Powers of that quantity, which is the Root. Thus, aa, aaa, aaaa, &c. are powers of the root a.

XIV. These powers are expreffed, by placing above the root, to the right hand, a figure, denoting how often the root is repeated. This figure is called an index or exponent, and from it the power is denominated. Thus,

The 2d and 3d powers are generally called the Square and Cube; and the 4th, 5th, and 6th, are also sometimes refpectively called the Biquadrate, Surfolid, and. Cubocube.

Cor. Powers of the fame root are multiplied by adding their exponents. Thus, a2×a3 a3, or aaa×aaĦaaaaa,b3×b=b+.

SCHO

SCHOLIUM.

Sometimes it is convenient to express the multiplication of quantities, by fetting them down with the fign. X between them, without performing the operation according to the preceding rules; thus, a2Xb is written instead of a2b; and a—bxc—d, expreffes the product of a-b, multiplied by c-d.

Def. XV. A Vinculum is a line drawn over any number of terms of a compound quantity, to denote those which are underftood to be affected by the particular fign connected with it.

Thus, in the last example, it fhews, that the terms a and b, and also c and ―d, are all affected by the fign X. Without the vinculum, the expreffion a-bxc-d, would mean the excess of a above bc and d; and a-bxc-d, would mean the excess of the product of a—b by c, above d. Thus alfo, a+b)2, expreffes the fecond power of a+b, or the product of that quantity multiplied by itself; whereas a+b2 would ex

prefs

prefs only the fum of a and b ; and so of others. By fome writers a parenthesis ( is used as a vinculum, and (a+b) is the fame thing as a+b)2.

PROB. IV. To divide Quantities. General Rule for the Signs. If the figns of divifor and dividend are like, the fign of the quotient is +; if they are unlike, the fign of the quotient is

This rule is eafily deduced from that given in Prob. 3.; for, from the nature of divifion, the quotient must be fuch a quantity as, multiplied by the divifor, fhall produce the dividend, with its proper fign.

From Def. 8. the Quotient of any two Quantities may be expreffed, by placing the dividend above a line, and the divifor below it. But a quotient may often be expreffed in a more fimple and convenient form, as will appear from the following diftinction of the Cafes.

CASE I. When the divifor is fimple, and is a factor of all the terms of the Dividend. This is easily discovered by inspection; for

then