is multiplied by x, and the product being equal to the dividend, the division is finished without any remainder. The operation may be proved, as before, by multiplying the quotient by the divisor, when the product will be equal to the dividend given in the question. It sometimes happens that the division will never come to a termination, without a remainder; in which case the quotient may be considered as infinite, and the rate of its progression may often be easily known; or the quotient may be brought to a conclusion in the shape of a fraction, of which the remainder is the numerator, and the divisor is the denominator. Example, divide 1 by 1-a. 1-a) 1 · (1 + a + a2 + a3 + aa + Here the 1 of the divisor being contained once in the 1 of the dividend, we place 1 in the quotient, and multiplying the whole divisor by 1, we obtain 1a, which sub tracted tracted from the given dividend, gives for a remainder + a for a new dividend, and the 1 of the divisor being contained a times in the dividend a, we write a with the proper sign prefixed in the quotient, and then muitiplying by it the whole divisor, the product is a -- a', which being subtracted from a, leaves +a; this being again divided by 1 of the divisor, the quotient is a, Ly which multiplying the whole divisor, we have a2 — a3, to be subtracted as before. In this manner the division may be carried on indefinitely, without ever coming to a termination but from the rate of progression it is evident, that the quotient will continually advance nearer and nearer to the truth, by an additional power of a ; the divisica, therefore, when this rate of progression is ascertained, may be intermitted, and the last remainder written, as a faction, at the end of the quotient, as in the example here given. With respect to the method pointed out for division of algebraic quantities, it may be observed, that, in the course of the operation, every term of the divisor being successively multiplied by every term of the quotient, and the several products subtracted from the given dividend, until nothing remain, or until the progression of the quotient be ascertained, the quotient thus obtained must be correct, as may be proved by multiplying it by the divisor, when the product, together with the remainder, if there will be equal to the given dividend. be any, In algebraic calculations two operations frequently occur, viz. Involution and Evolution. Involution means the way to discover any power of any given quantity, whether simple or compound, and is performed by multiplication. 1st. Involution of a simple quantity is performed by multiplying the exponents of the letters by the index of the power power required, and raising the co-efficient to the same power. Example, raise 2 a2 m3 to the cube, or 3d power: multiply 2, the exponent of a, by 3, denoting the power, making a; and 3, the exponent of m3, making mo; and cubing the co-efficient 2, thus making it 8, the 3d power of the given quantity 2 a2 m3, becomes 8 a mo. If the same be performed by multiplication, as here shown, the result will be the same. Raising the co-efficient 3 to the 6th power, we have a new co-efficient, 729; multiplying the exponent of m2 by 6, expressing the power required, we have m12; and multiplying the exponent of x3 also by 6, we have x1; consequently the 6th power of 3 m2 3 will be 729 m2 x18; as the student may discover by multiplying the quantity given 6 times into itself. 2nd. When the given quantity is composed of two or more terms, the required power must be found by successive multiplications of the quantity into itself; thus, for instance, to find the 4th power of a+m. a2 + 2am + m2 = 2d power or square a + m a3 + 2a2m— am2 a2m + 2am2 + m3 a3 + 3a2m + 3am2 + m3 3d power or cube a2 + 3a3m + 3a2m2 + am3 a3m + 3a2m2 + 3am3 + m4 a4 + 4a3m + 6a2m2 + 4am3 + m2 = 4th power required. Evolution is the method of discovering the root of any given quantity, simple or compound; its operations are, therefore, the reverse of involution. In algebra, to denote the root of any quantity, the radical sign (√) is used, with a figure over the quantity, to denominate the root required: thus √x is the square root of the 3 2 5 quantity expressed by x, √m is the cube root of m, √am is the 5th root of the quantity am. The figure is called the exponent or index of the root; and when the square or 2nd root is meant, this index is frequently omitted, so that vz and √x, represent the same square root. 2 1st. When the quantity of which the root is required is simple, divide the exponents of the letters by the index of the root, and prefix the root of the co-efficient to the letters, and this new found quantity will be the root required. Example, required the square root of 64 a2m2. Here the square root of the co-efficient 64 being 8, and the exponents 2 being divided by the index of the square or 2nd root, which is also 2, the result is Sam: and if this root be squared the product will be 64 a^m2. It is to be observed, that the root of any positive quantity may be either positive or negative, if the index of the root be an even number; for if + x and - be both squared, the product will still be the same, or +2; and the root of 2 would be expressed thus: +; but if the index be an odd number, the root will always be positive: the root of a negative quantity is always negative when the index of that root is an odd number; but if the quantity be negative, and the index an even number, no root can be assigned: thus no square root can be assigned for x2, for both + x and x if squared would give + x2. 2dly. When the quantity, the root of which is to be extracted, is composed of more than one term, if it be a square number, proceed as in the following example, where the square root of a2 + 2 a c + c2 is required. a2 + 2 a c + c2(a + c the root, Find the square root of the first term a2, which is a; place this root in the quotient, and its square under the dividend as there is no remainder, bring down the next period 2 a c + c2, and doubling the quotient, we have for a new divisor 2a; then enquiring how often 2 a can be had in 2 ac, the number of times c is placed in the quotient with the sign + prefixed, both divisor and dividend having the same sign, and also in the divisor, the whole of which multiplied by e gives 2ac + c', equal to the last dividend consequently the square root of a2+2ac + c2 is a + c. : The same operation may be performed with arithmetical 25 figures |