When the Quantities are Unlike. HAVING collected together all the like quantities, as in the two foregoing cases, set down those that are unlike, one after another, with their proper signs. Add a+b and 3a - 56 together. Add 6x-5b+a+8 to 5a-4x+4b. - 3. - Add a+26-3c-10 to 3b-4a+5c+ 10 and 56-c. Add 3a +b-10 to, c-d-a and -4c+2a-3b-7. Add 3a2+b2-c to 2ab-3a2+bc-b. Add a3+b3c-b2 to ab2-abc+b2. Add 9a-8b+10x - 6d-7c+ 50 to 2x-3a-5c +4b+6d -10. SUBTRACTION. SET down in one line the first quantities from which the subtraction is to be made; and underneath them place all the other quantities composing the subtrahend; ranging the like quantities under each other, as in Addition. Then change all the signs (+ and −) of the lower line, or conceive them to be changed; after which, collect all the terms together as in the cases of Addition*. * This rule is founded on the consideration, that addition and subtraction are opposite to each other in their nature and operation, as are the signs and, by which they are expressed and represented. So that, since to unite a negative quantity with a positive one of the same kind, has the effect of diminishing it, or subducting an equal positive 8xy-361-y From 8x2y+6 5/xy+2x/xy 7x2+2x-18+3b Take-2xy +2 7/xy + 3—2xy 9x2-12 +56+x From 4a+4b, take b + a. From 4a 4b, take 3a + 5b. From 8a - 12x, take 4a-3x. From 2x 4a-2b+5, take 8 — 5b + a + 6x. -d --- From a3+362c + ab2 — abe, take b2 + ab2 - abc. -40. From 6a -4b12c+12r, take 2a -8a4b5c. one from it, therefore to subtract a positive (which is the opposite of uniting or adding) is to add the equal negative quantity. In like manner, to subtract a negative quantity, is the same in effect as to add or So that, changing the sign of a quantity unite an equal positive one. from+to, or from to, changes its nature from a subductive quantity to an additive one; and any quantity is in effect subtracted, by barely changing its sign. MULTIPLICATION. This consists of several cases, according as the factors are simple or compound quantities. CASE 1. When both the Factors are Simple Quantities. FIRST multiply the co-efficients of the two terms together, then to the product annex all the letters in those terms, which will give the whole product required. Note*. Like signs, in the factors, produce +, and unlike signs, in the products. *That this rule for the signs is true, may be thus shown. 1. When a is to be multiplied by +c; the meaning is, that + a is to be taken as many times as there are units in c; and since the sum of any number of positive terms is positive, it follows that +ax+c makes+ac. 2. When two quantities are to be multiplied together, the result will be exactly the same, in whatever order they are placed; for a times c is the same as c times a, and therefore, when -a is to be multiplied by +, or+c by-a: this is the same thing as taking a as many times as there are units in+c; and as the sum of any number of negative terms is negative, it follows that - a ×+c, or+aXc make or pro duce -- ac. 3. Whena is to be multiplied by-c: here a is to be subtracted as often as there are units in c: but subtracting negatives is the same thing as adding affirmatives, by the demonstration of the rule for subtraction; consequently the product is c times a, or+ac. Otherwise. Since a―a = 0, therefore (aa) X-c is also = 0, because 0 multiplied by any quantity, is still but 0; and since the first term of the product, or a X-c is = - ac, by the second case; therefore the last term of the product, or ac, must be + ac, to make the sum = 0, or acac 0; that is, — a X- - c = + ac. Other demonstrations upon the principles of proportion, or by means of geometrical diagrams, have also been given; but the above may suffice. |