To Divide one Fractional Quantity by another. DIVIDE the numerators by each other, and the denominators by each other, if they will exactly divide. But, if not, then invert the terms of the divisor, and multiply by it exactly as in multiplication. EXAMPLES. 1. If the fractions to be divided have a common denominator, take the numerator of the dividend for a new numerator, and the numerator of the divisor for the new denominator. 2. When a fraction is to be divided by any quantity, it is the same thing whether the numerator be divided by it, or the denominator multiplied by it. 3. When CASE II. When the Quantities are Like, but have Unlike Signs; ADD all the affirmative co-efficients into one sum, and all the negative ones into another, when there are several of a kind. Then subtract the less sum, or the less co-efficient, from the greater, and to the remainder prefix the sign of the greater, and subjoin the common quantity or letters. INVOLUTION. INVOLUTION is the raising of powers from any proposed root; such as finding the square, cube, biquadrate, &c, of any given quantity. The method is as follows: MULTIPLY the root or given quantity by itself, as many times as there are units in the index less one, and the last product will be the power required.-Or, in literals, multiply the index of the root by the index of the power, and the result will be the power, the same as before. Note. When the sign of the root is +, all the powers of it will be+; but when the sign is -, all the even powers will be +, and all the odd powers; as is evident from multiplication. Any power of the product of two or more quantities, is equal to the same power of each of the factors, multiplied together. And any power of a fraction, is equal to the same power of the numerator, divided by the like power of the denominator. Also, powers or roots of the same quantity, are multiplied by one another, by adding their exponents; or divided, by subtracting their exponents. Thus, a3 X a2 = a 3+2 = q3. And a agor 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*. 7x2 + 2 √ x − 18+ 36 +56 + x 2 From 8x2y+6 5√xy + 2x √xy Rem. 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 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 unite an equal positive one. So that, by changing the sign of a quantity 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. the same height as the intended power: and the last term of the series will contain only the 2d part of the given root, when raised also to the same height of the intended power: but all the other or intermediate terms will contain the products of some powers of both the members of the root, in such sort, that the powers or indices of the 1st or leading member will always decrease by 1, while those of the 2d member always increase by . 2. To find the Co-efficients. The first co-efficient is always 1, and the second is the same as the index of the intended power; to find the 3d co-efficient, multiply that of the 2d term by the index of the leading letter in the same term, and divide the product by 2; and so on, that is, multiply the coefficient of the term last found by the index of the leading quantity in that term, and divide the product by the number of terms to that place, and it will give the co-efficient of the term next following; which rule will find all the co-efficients, one after another. Note. The whole number of terms will be 1 more than the index of the given power: and when both terms of the root are, all the terms of the power will be +; but if the second term be, all the odd terms will be +, and all the even terms —, which causes the terms to be + and alternately. Also the sum of the two indices, in each term, is always the same number, viz. the index of the required power and counting from the middle of the series, both ways, or towards the right and left, the indices of the two terms are the same figures at equal distances, but mutually changed places. Moreover, the co-efficients are the same numbers at equal distances from the middle of the series, towards the right and left; so by whatever numbers the increase to the middle, by the same in the reverse order they decrease to the end. EXAMPLES. 1. Let a +x be involved to the 5th power. The terms without the co-efficients, by the 1st rule, will be a3, a1x, a3 x2, a2x3, ax1, x3, and the co-efficients, by the 2d rule, will be 5 X 4 10 X 3 10 X 2 5 x 1 or 1, 5, 10, 10, 5, 1; Therefore the 5th power altogether is as +5α*x + 10a3x2 + 10a2x3 + 5ax♦ + x3. 'VOL. I. Dd But, |