26. Involution is performed by the fucceffive multiplication of any quantity into itfelf. A quantity multiplied into itself produces the fquare of the fame; and the fquare multiplied again by the original quantity produces the cube of the fame; and that cube again multiplied by the root gives the biquadratic power. And so on. 27. Simple quantities are involved by multiplying their exponents by that of the power, and prefixing a like power of the coefficient. Thus, the fquare of b is b2; the cube of da is 512a3. 28. Pofitive roots give pofitive powers; but negative roots give pofitive and negative powers by turns. 29. Any two quantities connected by the fign +, are called a Binomial; but if connected by the fign, a Residual. EXAMPLE I. Required the fquare, cube, biquadratic, furfolid, and fixth power of a+b. a+b=root. a+b a2+ab +ab+62 a2+2ab+b2=square. a+b a3+2a2b+ab2 +a2b+2ab2+b3 a3+3a2b+3ab2+63=cube. a+b a*+3a3b+3aRb2+ab3 + a3b+3a2b2+3ab3+b4 a++4a3b+6a2b2+4ab3+b*=biquadratic. a+b a2+4a4b+6a b2+4a2b3+aba +a+b+4a3b2+6a2b3+4ab4+bs a3+5aab+10a3b2+10a2b3+5ab4+b=fursolid. a tb a°+5a3b+10a+b2+10a3b3+ 5a2b4+ abs a3b+5a+b2+10a3b3+10a2b4+5abs +bo a°+6a3b+15aab2+20a3b3+15a2b4+6ab3 +bo=6th power. EXAMPLE EXAMPLE II. equired the fquare, cube, and biquadratic powers of a-b. 30. It appears by reviewing these examples, that all the terms of the powers of a binomial are pofitive; but the terms of the powers of a refidual are pofitive and negative alternately, the first pofitive, the second negative; the third pofitive, the fourth negative; and fo on, and by turns. Also, that the sum of the exponents of a and b, in any of the intermediate terms, is equal to the exponent of the first or laft term; and that the exponent of the first or of the last term is equal to that of the power. In the first term b is wanting, and the power of a in every fucceeding term decreases regularly by 1; and that of b encreases in each term by 1, until a dif appear. The coefficient of the first term is 1: The coefficient of the fecond term is equal to the exponent of the first. One or more terms being found, the cofficient of the next fucceeding term may be discovered in this manner: Multiply the exponent of a in the last term by the coefficient of the fame; divide the product by the number of terms already made up, and the quot will be the coefficient required. 31. From thefe obfervations we may infer the following rule, commonly called the a+b, or the Binomial Theorem, by which we may involve either a binomial or a residual root to a power of any dimenfion. RULE ift, To find the first term of the power. Multiply the exponent of a in the root by that of the power, for the first term of the power required. 2d, To find the fecond term of the power. Multiply the exponent of a in the firft term by the coefficient of the fame, divide the product by the number of terms alrea dy found the quotient will be the coefficient of the fecond ⚫ term of the power; then diminish the exponent of and encrease the exponent of b, each by 1, for the fecond term. 3d, To find the third term of the power. a, Multiply the exponent of a in the fecond term by the coefficient of the same divide the product by 2, (the number of terms already found); the quotient gives the coefficient of the third term; then decrease the exponent of a in the fecond term, and encrease that of 5 in the fame, each by unity, for the third term. 3 F 4th, 4th, To find the fourth term. Multiply the exponent of a in the third term by the coefficient of the fame; divide the product by 3, (the number of terms already found); the quotient is the coefficient of the next term; then take off a power of a and bring on a power of ¿, for the fourth term of the power complete: Continue this procefs till all the powers of a are exhaufted, and the power of b be equal to that of the power required. EVOLUTION. 32. Evolution is the operation by which roots are discovered, and is always oppofed to Involution. Roots are quantities by whose successive multiplication given powers are produced. 33. The roots of fimple quantities are extracted, by dividing the exponent of the power by the exponent of the root required. Thus, the cube root of a3 is a, of 86° is 2b1. The reafon of this is deduced from § 27. 34. Rules for extracting roots of compound quantities are deduced from a review of the fteps by which they are involved. Thus, the fquare of a+b is a2+2ab+b2; that is, the fquare of any two quantities is equal to the fquares of each of the quantities, together with twice their product. See Euclid, Book 2d, propofition 4th. Therefore when a quantity is proposed, whose fquare root is a compound quantity, you are firft to arrange the terms as taught in divifion, (§ 22.) Thus, Let the fquare root of a'+2ab+b2 be required. a2+2ab+b2 (a+b a2 2a+b)2ab+b2 We |