3 a 23. Add the sum, difference, product, and quotient, of 4 52. To involve simple quantities to any power. RULE I. Involve the coefficient to the power required, for a coefficient. II. Multiply the index of each letter by the index of the required power. b Involution being nothing more than continued multiplication, it will be sufficient to refer the reader to the notes on multiplication, where he will find the reason of this rule. The Arabians denominated the powers from the consideration of the products of the indices, calling them the square, cube, biquadrate, sursolid, cube-squared, second-sursolid, quadrato-quadrato-quadratum, cube-of-the-cube, square-of-thesursolid, third-sursolid, &c. Diophantus, Vieta, Oughtred, and others, named the powers according to the sums of the indices, as root, square, cube, quadrato-quadratum, quadratocuhus, cubo-cubus, quadrato-quadrato-cubus, quadrato-cubo-cubus, cubo-cubocubus, &c. Des Cartes, and most of the writers since his time, employ a much simpler method than either of the former, calling the powers respectively, the 1st, 2nd, 3rd, 4th, &c. according to the index. And because the side of a square multiplied into itself gives the area of the square, and the side of a cube multiplied continually twice into itself produces the solidity of the cube, the terms square and cube have been applied to numbers arising from like operations. Hence it is that the product of a number multiplied into itself is called a square; and if the number be multiplied twice into itself, the product is called a cube. It is evident that all the powers of an affirmative quantity will be affirmative, for into always produces +; likewise that all the even powers of a negative quantity will be +, and the odd powers -; for since — into produces, and this into produces, and this into produces, and -; it follows that the even powers will be, and so on alternately and the odd powers -. - III. Place each product over its respective letter, and prefix the coefficient found above, the result will be the power required. IV. Of an affirmative quantity all the powers will be + ; and of a negative quantity the odd powers will be —, and the even powers +. 2 to the second and third powers. Ans. second third power 5. Involve xy223 to the third power. Ans. x3y6z9. RULE. Multiply the given quantity as many times continually into itself wanting one, as there are units in the index of the required power, and the last product will be the power required. This rule is merely multiplication, and depends on the same principles, 7. Involve a+ and a−x, each to the cube. I multiply a + and a-r each by itself, and the product is the square; I multiply this by a+r in one, and a-x in the other, and the product in each is the cube: it will be seen that these operations differ only in the signs of their even terms. x1+3x3≈+3x2 z2 + xz3 + x3z+3x2 z2 + 3 xz3 +z+ x2 + 4x3 z+6 x2 z2 + 4 xz3 + z1 fourth power. +x+z+ 4x3 z2 + 6 x2 z3 + 4xz++z5 x3±5 x1z+10 x3 z 2 + 10 x 2 z 3 +5 xz*+z5 fifth power. 9. Involve +1 to the cube. Ans. x3+3x + 3x + 1. Ans. 8 x3-36 x2y+54 xy2 10. Involve 2x-3y to the cube. -27 y3. 11. Involve 3 - to the fourth power. Ans. 81—108x+54x2 --12 x3 + x*. 12. Involve a+b-c to the square. Ans. a2+2 ab—2 ac+b2 -2 bc+c2. SIR ISAAC NEWTON'S RULE FOR INVOLUTION'. Whereby any power of a given compound quantity may be obtained by an easy and expeditious mental operation. 54. For Binomials. To find the terms and indices. RULE 1. Write down the leading quantity successively, as many times as there are units in the index of the required power. II. Over the first of these place the index of the power; over the second, the index decreased by 1; over the third, the index decreased by 2; and so on, making the index of each term always 1 less than that of the preceding term. III. Subjoin the following quantity to the second, and every succeeding term, of the above, and carry it one place beyond. IV. Make 1 (understood) the index of the first of these; 2 the index of the second; 3 of the third, and so on, constantly increasing by 1 to the last; the index of which will be that of the required power. Thus, if it be required to involve a-z to the fifth power, the quantities and indices will stand as follows. The leading quantity a, thus, a, aa, a3, Both quantities connected... a3, a1z, a2z2, a2z3, az1, z3. To find the coefficients of the terms. RULE I. The coefficient of the first term is always 1, (under. stood,) and that of the second term is always the index of the required power. d This rule is a branch of the celebrated Binomial Theorem, discovered by Sir Isaac Newton in 1669. The author first discovered it by induction, namely, by observing the law which the signs, coefficients, and indices invariably follow, in a Binomial actually involved to several different powers. It will be a profitable employ for the learner to make the same induction: let him compare the 8th example with Newton's rule, and he will see that the coefficients, indices, signs, &c. of the terms, in every one of the powers, observe invariably the law on which the rule is founded. |