CASE II. To add like quantities with unlike signs. Rule 1. Collect the positive co-efficients into one sum, and the egative ones into another. 2. From the greater of these sums subtract the less, and to the remainder prefix the sign of the greater, and annex the common letters. Note.--If the aggregate of the positive terms be equal to that of the Degative ones, their sums will be = 0. Eramples. 2 4 -31+ 4 - ab + 3 bc xy -2y+2 azt -5a' +13. + I-5 ab +2 bc + 4 xy + y + art -249-4.1% -- 5.3 + 1 3 ab- bc +2 xy -7y--3 axt 77 + +21-4 -2 ab + 4 bc 3 XY +5y+ 3 art gr-14. 47+13 3 ab-8 bc + XY -- y art --13. 2.r$ --Or+9 say, 50, or IUus. 1.–The manner in which we generally calculate a person's property, is an apt illustration of the foregoing Note, and conseqnently of the rule. We denote what a man really possesses by positive pumbers, using the sign +; whereas bis debts are represented by negative numbers, or by understanding the sign —, as affecting those numbers. 2. Thus, when it is said of any one that he has 100 crowns, but owes 50, this means that his real possession amounts to 100-50, or which is the same thing + 100-50, that is to And if he has in possession 20 crowns, but owes 20, bis real possession amounts to 20—20, or + 20-2050. In fine, he has nothing; but then he owes nothing. But, on the other band, if he owes 70 crowns, and has in possession only 40, his real possession would be expressed thuis 70 + 40. Here his debt is fairly represented by the negative number - 70, while his real possession is represented by the positive number + 40. It is certain, therefore, that he has 30 crowns less than pothing; and we mnight, consequently, express the state of his finances — 30 ; for if any one were to make him a present of 30 crowns to pay this debt, 30, he would only be at the point pothing ( 0 ), though really richer than when 30 stood against his future prospects and exertions. 3. Debts, or sums of money owing, are therefore as much real sums, or quantities, of money, or real pombers, as credits are ; and the sign +, or - gruerns the quantity or number that follows it. * Scholium.- In the language of Algebra, a and b may stand for any two bombers whatever (Notes 1 and 2, p. 1); and, therefore, at b stands for u made more by b. Again, a - b stands for a made less by b; that is, for the difference of a and 6 (where b is supposed less than a). Now, by the role of Case II. a + b and a b, added together, make twice a (2 a); therefore we derive this (25.) Theorem. If the sum and difference of any two numbers be added together, the whole will be twice the greater number; for if a + b be added to a -0, the sum is 2a. Case III. To add unlike quantities. Rule. Collect all the like quantities, by the last rules, and set down those which are unlike, one after another, with their proper signs. A Scholium is a remark or observation made on some foregoing роsition, or other premises. † A Theoren is a demonstrative proposition, in which some property is asserted and the truth of it required to be proved. - 8 y 1 -107 -3 I + ry -88 zy ry +9x2 Examples. 4 2r + vrty 4 y +/- 2 y VX + 100 - 5.x -8 + vry SUBTRACTION. (26.) Rule 1. Write, in one line, those quantities from which the subtraction is to be made, and which we call the minuend; then onderneath write all the quantities to be subtracted, which we call the subtrahend, ranging under each other the quantities of the same denomination. 2. Change the signs of the quantities to be subtracted, or conceive them changed; then collect the different terms, and place them as directed by the rules of Addition.* Examples. 1 2 3 4 ato 6 x2 8 y + 3 2 x + 5 5 y: - 4y + 3 a 2 x + 9 y 2 3 r + 5 x 7 x 6 ya 4 y Scholium.-In the scholium to Case II. of Addition, we shewed that a + b may represent the sum, and a - - b the difference of any two_oumbers, of which a is the greater and b the less. Now here it appears (in Example 1. of Subtraction), that if a - 6 be taken from a + by the remainder will be twice b (2 b); whence we derive this (27.) Theorem. If the difference of any tuo numbers le subtracted from their sum, the remainder will be twice the less number. 2 • This rule may be thus illustrated : If it were required to subtract 5 (i. e. 3) from 9, it is evident that the remainder would be greater by 4, than if 5 only were subtracted. For the same reason, if I were subtracted from a, the remainder would be greater by c than if b only were taken away. Now if + b be subtracted from t a, the remaiuder will be a - b; and consequently, if b c be subtracted from a, the remainder will be a b + c. If b were a negative quantity (b) to be taken from t a (or «), we should obtain a + b. For the same reason when c is a negative quantity (- c), and b a positive one, as in the expression just given, we change the signs of both, thus: -b+c, when we would take them from a. MULTIPLICATION. (28.) In the multiplication of algebraic quantities, four circumstances are to be considered. 1. The signs of the quantities : Their co-efficients : 3. The letters of which they are composed ; and 4. The indices or exponents of those letters. (29.) In performing any operation in multiplication, we must, therefore, ubserve the four following rules. 1. When quantities having like signs are multiplied together, the product will be to. On the contrary, if their signs are unlike, the sign of the product will be —* 2. That the co-efficients of the factors must be multiplied together, to form the co-efficient of the product. 3. That the letters of which the factors are composed must be set down, one after another, according to their order in the alphabet. 4. That if the same letter be found in both factors, the indices of this letter must be added, to forin its index in the product. or . That like signs make t, and unlike signs - in the product, may be il. lustrated thus : First. When + a is to be moltiplied by + 6, this denotes that + a is to be taken as many times as there are units in b; and because the sum of any number of affirmative terms is affirmative, it is obvious that to x + 6 = + ab. Secondly.--If two quantities are to be multiplied together, the result will be actually the same, in whatever order they are placed : for a times b is the same as b times a; and, therefore, wlien -a is to be multiplied by + b, or tb by 2, it is the same thing as taking -a as many times as there are units in to b; and as the sum of any number of negative terms is negative, it is plain that - ax + b, or +0 X-Q= ab. Lastly. When - a is to be multiplied by - b, we have ab for the product at first sight, but still we must determine whether the sign + or - is to be placed before the product. Now it cannot be the sign —, for + X -, which is the same thing, tax- b gives - ab, and a by - t cannot produce the same result as - ax + b; but must produce a contrary result, to wit, tab; consequently we have the following rule: multipdied by - produces +, in the same manner as + x + give te But this illustration may be demonstrated thus: When the compound quantity + amb is to be multiplied by + c, we repeat or add + a - b to itself as often as there are units in c; hence, since the sum of any number of affirmative terms is affirmative, and the sun of any number of negative terms is negative, it is obvious, that + a -- ó mul tiplied by + c produces + ac – be; for the same reason, ta- b multiplied bý + d produces + od -6. Wrenrc, if from times (a 6) + ac - bu al + «d like signs prodnce po! * be + be unlike s os produce minus. be a ........ (30.) From these four rules we have i. As to their signs ta x + a.. ...3 + al. 6. . , . Note. From the division of algebraic gnantities into simple and compound, three cases of multiplication arise ; and in performing the operation in all these cases, we must attend first to the signs, then the co-efficients, and lastly the letters and indices. Case I. When both the factors are simple quantities. Rule. Attend to the signs, co-efficients, and indices, by the foregoing Rules (29 and 30). Examples. 2 4 5 - 7 xyz 17 ta yo 5 ar - 2 ara 1 3 4 6 CASE II. When one factor is compound, and the other simple. Rule. Multiply each term of the compound factor by the simple factor, as in the last case, and the result will be the product required. Examples. 1 2 3 3ab 2ac + d 3rd 2 x3 + 4 771-4X +4 0 4a 12 ax 30 Case III. When both factors are compound quantities. 1 Rule. Multiply each term of the multiplicand by each term of |