5 ac ab 7 ad 5 xy From the four preceding examples, it may be easily perceived, that fubtraction in algebra is proved as in common arithmetic, by adding the remainder to the qnantity which is fubtracted. E. 9. From E. 8. 5232 am 3 zy + 4 am Remains 6 am 17 a 827 If the quantities to be fubtracted are unlike those from which the fubtraction is to be made, fet down these with the fame figns and coefficients they have in the example; after which, place the quantities to be fubtracted with their coefficients, but change their figns. From E. 10. bd Remains 3ac-bd In Example 10. having put down quantity to be fubtracted being + bd, required. MULTIPLICATION. In Multiplication there is one general rule for the figns, viz. when the figns of the factors are alike (that is, both-or both) the fign of the product is+; but when the figns of the factors are unlike, the fign of the product is. This general rule will refolve itfelf into four partis cular cafes, which I fhall illuftrate feparately in fimple quantities. CASE I. When any pofitive quantity, as+a, is multiplied by a pofitive quantity+b, the meaning is, that a is to be taken fo many times as there are units in b, and the product is evidently 6 times aj or ba. In example 1, having joined the letters ba, and each of them having the affirmative fign, therefore, by the rule, ba or+ba is the product required and fo of others. is multiplied by b, then CASE 2. When a -a is to be taken as often as there are units in b, and the product must be b times, orba. EXAMPLES. 7bx -27dc - In example 1, cafe 2, the product of a by b is ba, and as the fign of a is-> and that of b is, therefore to b a prefix the fign —, fo is-ba the product required. 1. TSL CASE 3As multiplication by a pofitives number implies a repeated addition, fo multiplication by a negative implies a repeated fubtraction; and therefore, when a orta is to be multiplied by -b, it means only, thata is to be fubtracted as often as there are units in b, and therefore the product being negative, must also be-ba; fee the following ba 12ab -48 b c -36acdx CASE 4 When- a is to be multiplied by b, then fubtracted as often as there are units in ; but to fubtract valent to adding+a; therefore this cafe is the fame in effect as cafe 1, and the product is evidently+ba, or ba. a is to be axis equi EXAMPLES. A compound quantity is multiplied by a fimple one, by multiplying every term of the multiplicand by the multiplier. -ab+bb-bc If there are coefficients, or numbers prefixed to the quantity, then multiply the numbers as in common arithmetic, and to their products join the products of the quantities found by the laft example. ́4a3 b c — 8 a2b2c+6a2bca—4abac2+6āb3c→→→4abc3 If any algebraic quantities are to be multiplied by a pure number, this number is to be multiplied into every one of the coefficients of the the other quantities, in all refpects as before, and to each particular product fet or join that quantity whose coefficient was multiplied. EXAMPLES. Multiply 3a+46 Compound quantities are multiplied into one another, by multiplying every term of the multiplicand by each term of the multiplier, fucceffively, and connecting the feveral products thus arifing, with the figns of the multiplicand, if the multiplying term be affirmative, but with contrary figns, if negative. In the above example, by ftriking out all the terms that deftroy each other, the product becomes aa-64. pro Note. If the fign of any propofed term of the multiplier, in any cafe whatever, be affirmative, it is eafy to conceive, that the required product will be greater than it would be if there was no fuch term, by the duct of that term into the whole multiplicand; and therefore it is, that this product is to be added or wrote down with its proper figns. But if, on the contrary, the fign of the term by which you multiply be negative, then, as the required product must be lefs than it would be, if there were no fuch term, by the product of that term into the whole multiplicand, this product, it is manifeft, ought to be subtracted or wrote down with contrary figns. Hence is derived the common rule, that like figns produce +, and unlike figns For, firft, if the figns of both the quantities or terms to be multiplied are affirmative, it is plain, that the fign of the product muft likewife be affirmative. Secondly, if the figns of both quantities are negative, that of the product will be affirmative, because contrary to that of the multiplicand, as proved above. Thirdly, if the fign of the multiplicand be affirmative, and that of the mulitplier negative, the fign of the product will be negative, because the fame with that of the multiplicand. Laftly, if the fign of the multiplicand be negative, and that of the multiplier affirmative, the fign of the product will be negative, because the fame with that of the multiplicand. And these are all the cafes that can poffibly happen with regard to the variation of figns. 30 Examples a3—3a+b+3a3b2—a2b3 --2a+b+6a3b2 6a2b3+2ab+ Product as-5a+b+10a3b2 —10a2b3+5ab*—b$ DIVISION. In divifion of algebraic quantities, the rule for the figns is the fame as in multiplication, viz. if the figns of the divifor and dividend are alike, the fign of the quotient must be + but if they are unlike, the fign of the quotient must be This is a general rule for all operations in divifion, which are only the reverse of multiplication, and therefore will be easy to understand, when illuftrated by examples. Quotient d EXAMPLES. -mad a In the first example, because ac is in the dividend and divifor, reject it, and put downd for the quotient. The truth of thefe operations in divifion may be proved like those in arithmetic; for the quotient and divifor being multiplied, the product will be the dividend, if the work is true; thus, in the fecond example, by multiplying a the quotient into md the divifor, the product is mda, or adm, or mad, to which must be prefixed the fign, because the figns of md and a are unlike; hence the product with its fign is mad, the given dividend. The truth of these examples are proved as in common arithmetic. E. 8. a+x}a3+5a2x+5@x2+x3{a2+4x+x2 4a2x+5ax2 ex2+x3 ex2+x3 ** To work the above example, fay, how often is a contained in a3 the answer is 42, which I write down in the quotient, and multiply the whole divifor ax thereby, and there arifes a3a2x; which, fubtracted from the two first terms of the dividend, leaves 4a2x; to this remainder I bring down+sex, the next term of the dividend, and then feek again how many times a is contained in 4022; the answer is 47x, which I alfo put down in the quotient, and by it multiply the whole divifor, and there arifes 44x+4x2, which fubtracted from 442 +54x2, leaves ax210 which I bring down x3, the laft term of the dividend, and feek how many times a is contained in 2x2, which I find to be x2, which I also put in the quotient, and by it multiply the whole divifor, and then, having fubtracted the prodect from ex2+-x3, I find there is nothing remains, E. 9x-3x2-x3 a2—2«x+x2}«3—5e*x+10@3r2—104*x3†5«x*—x3 {e3—3aa 302 |