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 multiplying by it the whole divisor, the product is a- a', which being subtracted from a, leaves + a2; this being again divided by 1 of the divisor, the quotient is a', by which multiplying the whole divisor, we have a 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 division, therefore, when this rate of progression is ascertained, may be intermitted, and the last remainder written, as a fraction, 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 be any, will be equal to the given dividend.

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 m9; and cubing the co-efficient 2, thus making it 8, the 3d power of the given quantity 2 a2 m3, becomes 8am. 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 m3 by 6, expressing the power required, we have m12; and multiplying the exponent of x3 also by 6, we have x1s; consequently the 6th 3 m2 x3 will be 729 m2 x18; as the student may discover by multiplying the quantity given 6 times into itself.

of power

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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.

a + m = 1st power or root

a + m

a2 + am
am + m2

a2 + 2am + m2 = 2d power or square

a + m

a3 + 2a2m— am3

a2m + 2am2 + m3

a3 + 3a2m + 3am2 + m3 = 3d power or cube

a + m

a4 + 3a3m + 3a2m2 + am3

a3m+3a2m2 + 3am3 + m2

aa + 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 is the square root of the

quantity expressed by x, vm 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 √ 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 a3m2.

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 x be both squared, the product will still be the same, or + x2; and the root of 2 would be expressed thus: +x; 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 for2, 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 a + 2 a c + c2 is required.

a2 + 2 a c + c2(a + c the root,

Find the square root of the first term a', 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 ac+c, 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 2ace, equal to the last dividend consequently the square root of a2+2ac + c2 is

a+c.

The same operation may be performed with arithmetical

2 U

figures

figures as follows, supposing the quantity represented by a to be 8, and that represented by c to be 5.

Extraction of the cube root is performed as in the foflowing example, where it is required to find the cube root of a3 + 3 a' c + 3 a c2 + c3.

a3 + 3 a2c + 3ac2 + c3 (a + c cube root.

Find first the cube root of a3, which is a; this being cubed, the product is placed under a3 of the given cubic quantity, and being subtracted from the whole quantity, leaves the remainder 3 ac + 3 ac2+ c'; then taking three times the square of the quotient or root a, we have for the first term of the divisor 3 a2, by which dividing the first term of the dividend 3 a c, we have for a second term in the root + c, by which dividing the other terms of the given quantity 3 a c2+c2, we have 3 a + c2 for the other terms of the divisor;