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 résult will be the power, the same as before. of all the even powers Note. When the sign of the root is +, all the powers it will be ; but when the sign is will be +, and all the odd powers multiplication. ; as is evident from a2, the root a4 = square a = cube 8 a = 4th power 3ab2, the root +9a2b=square 27a3b6 cube +81a4b84th power. -243a5b5th power. power a the root 26' * 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 xa2 = a3 +2 = α2. And a3 a or a = a. EXAMPLES FOR PRACTICE. 1. Required the cube or 3d power of 3aa. 4. To find the biquadrate of a2x 262 SIR ISAAC NEWTON'S RULE for raising a Binomial to any Power whatever *. 1. To find the Terms without the Co-efficients. The index of the first, or leading quantity, begins with the index of the given power, and in the succeeding terms decreases continually by 1, in every term to the last; and in the 2d or following quantity, the indices of the terms are 0, 1, 2, 3, 4, &c, increasing always by 1. That is, the first term will contain only the 1st part of the root with the same index, or of *This rule, expressed in general terms, is as follows: Note. The sum of the co-efficients, in every power, is equal to the number 2, when raised to that power. Thus 1+1=2 in the first power; 1 + 2 + 1 = 4= 22 in the square; +18 23 in the cube, or third power; and so on. +3+3 the 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 1. 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 + be involved to the 5th power. The terms without the co-efficients, by the 1st rule, will be a3, a1x, a3x2, a2x3, «x1, x3‚ and the co-efficients, by the 2d rule, will be Therefore the 5th power altogether is as + 5a x + 10 r2 + 10a2x23 + 5ax+ + x3. But But it is best to set down both the co-efficients and the powers of the letters at once, in one line, without the intermediate lines in the above example, as in the example here below. 2. Let a -r be involved to the 6th power. The terms with the co-efficients will be ao—6a3x + 15a2x2-20a3x3 + 15a2xa—6ax2 + x3. 3. Required the 4th power of a-x. any Ans. a1 — 4a3x + 6a2x2 — 4ax3 + x*. And thus other powers may be set down at once, in the same manner; which is the best way. EVOLUTION. EVOLUTION is the reverse of Involution, being the method of finding the square root, cube root, &c, of any given quantity, whether simple or compound. CASE I. To find the Roots of Simple Quantities. EXTRACT the root of the co-efficient, for the numeral part; and divide the index of the letter or letters, by the index of the power, and it will give the root of the literal part; then annex this to the former, for the whole root sought*. * Any even root of an affirmative quantity, may be either + or thus the square root of a2 is either +a, ora; because + a + a = + a2, and -ax a=a2 also. But an odd root of any quantity will have the same sign as the quantity itself: thus the cube root of + a3 is + a, and the cube root of - a3 is — a ; for + a x + a x + a = + a3, and — a × Any even root of a negative quantity is impossible; for neither +ax+a, nor —a × a can produce a2. Any root of a product, is equal to the like root of each of the factors multiplied together. And for the root of a fraction, take the root of the numerator, and the root of the denominator. EXAMPLES. To find the Square Root of a Compound Quantity. THIS is performed like as in numbers, thus : 1. Range the quantities according to the dimensions of one of the letters, and set the root of the first term in the quotient. 1 2. Subtract the square of the root, thus found, from the first term, and bring down the next two terms to the remainder for a dividend; and take double the root for a divisor. 3. Divide the dividend by the divisor, and annex the result both to the quotient and to the divisor. 4. Multiply the divisor, thus increased, by the term last set in the quotient, and subtract the product from the dividend. And so on, always the same, as in common arithmetic. EXAMPLES. 1. Extract the square root of aa- 4a3b + 6a2b2 — 4ab3. + ba. aa— 4a3b + 6a2b3 — 4ab3 + ba ( a2 - 2ab + b2 the root. |