| XXX | (81) | XXX |
Case ^11, To add quantities that are unlike. Rule. Set them all down one. after another, with their figns and coefficients prefixed. Exa MPLE. To '+ 2* I + 3< Add + 35 —4* Sum 2 « + 3 £ Chap. II. Of Subtraction. General Rule. “ Change the figns of the quantity to be fubtrafted into their contrary figns, and then add it fo changed to the quantity from which it was to be fubtrafted, (by the rules of the laft chapter) : the fum arifing by this addition is the remainder.” For, to fubtraft any quantity, either pofitive or negative, is the fame as to add the oppofite kind. Examp. From + 54 8a — q b Subtraft -j-ga. 3a+45 Remaind. $ a —^ a, or 2 «. S a — 11 b It is evident, that to fubtraft or take away a decrement is the fame as adding an equal increment. If we take away — b from a — b, there remains a; and if we add + b fo a — b, the fum is like^ife a. In general, the fubtraftion of a negative quantity is equivalent to adding its pofitive value. Chap. III. O/^ Multiplication. In Mutiplicatiojn, the general rule for the figns is. That when the figns of the fatlors are like, (i. e. both -F, or both —), the fign of the product is 4-; but when the figns of the faftors are unlike, the fgn of the produd is —. Case I. When any pofitive quantity, 4-- a, is multiplied by, any pofitive number, ft n, the meanis. That 4- a is to be taken as many times as there are units in n; and the produft is evidently n a. Case II. When — a is multiplied by «, then — a is to be taken as often as there are units in n, and the produft muft be — » a. Case III. Multiplication by a pofitive number implies a repeated addition: But multiplication by a negative implies a repeated fubtraftion. And whep 4- a is to be multiplied by — », the meaning is, That 4“ « is to be fubtrafted as often as there are units in «: Therefore the produft is negative, being ■— n a. Case IV. When — a is to be multiplied by —n, then - ■— a is to be fubtrafted as often as there are units in n ; but (by chap. II.) to fubtraft — a is equivalent to adding 4* a, confequemly the produft is -J- « 4. Vox. I. No.'4. 3
A. R Th.e lid and IVth Cafes may be illuftrafed in the following manner. By the definitions, -- a —■ a ~ o; therefore if we multiply ^* a —■ a by «, the produft: muft vaniffi, or be o, becaufe the faftor a — a is o. The firft term of the produft is 11 a (by Cafe I.) Therefore the fecond term of the produft rauft be — n a, which deftroys 4- /? a ; fo that the’ whole produft: muft be 4* « « — « a — o. Therefore — a multiplied by 4" « gives — n a. In like manner, if we multiply a — a by— n, the firft term of the produft being — n a, the latter term of the produft muft be 4" » ^; becaufe the two toge-: ther muftdeftroy each other, or their amount be o, fince one of the faftors (viz. a •— d) is o. Therefore — a multiplied by — n, muft give + « a. In this general doftrine, the multiplicator is always confidered as a number. A quantity of any kind may1 be multiplied by a number. If the quantities to be multiplied are fmple quantities, “ find the fign of the produft by the laft rule; af“ ter it place the produft of the coefficients, and then “ Tet down all the letters after one another as in one “ word.” Examp. Mult. a I —- 2 « 6x By 4- * I + 4 * 5a Prod. + ab Mult. g x 4- 3* By — 4 a — so Prod. 4-32 ax —lyaa^t To multiply compound quantities, you muft “ multiply ‘ “ every part of the multiplicand by all the parts of the “ multiplier, taken one after another, and then colleft ’ “ all the produfts into one fum: That fum ffiall be the . “ produft required.” Examp. Mult, ab i — 35 By a + b ' + S* ^8aa — 12 a b Prod,'•{"is b-trbb 4- xoab — Sbb Sum a a 2 a b-- bb 8 a a — 2 a Mult, a a + a b --b b By a — b -^abb Prod. —aa bb-abb — bbb Produfts that arife from the multiplication of two, three, dr more quantities, as a b c, are faid to be of two, three, Or more dimenfions; and thofe quantities are called faftars or roots. If all the faftors are equal, then thefe produfts are called powers ; as a a, or a a a, are powers of a. Powers are exprelfed ■ fometimes by placing above the X root,
