* To Subtract one Fractional Quantity from another.

REDUCE the fractions to a common denominator, as in addition, if they have not a common denominator.

Subtract the numerators from each other, and under their difference set the common denominator, and the work is done.

* In the addition of mixed quantities, it is best to bring the fractional parts only to a common denominator, and to annex their sum to the sum of the integers, with the proper sign. And the same rule may be observed for mixed quantities in subtraction also. EXAMPLES.

To Multiply Fractional Quantities together.

MULTIPLY the numerators together for a new numerator, and the denominators for a new denominator.*

1. When the numerator of one fraction, and the denominator of the other, can be divided by some quantity, which is common to both, the quotients may be used instead of them.

2. When a fraction is to be multiplied by an integer, the product is found either by multiplying the numerator, or dividing the denominator by it; and if the integer be the same with the denominator, the numerator may be taken for the product.

EXAMPLES.

To Divide one Fractional Quantity by another.

DIVIDE the numerators by each other, and the denominators by each other, if they will exactly divide. But, if not, then invert the terms of the divisor, and multiply by it exactly as in multiplication. EXAMPLES.

1. If the fractions to be divided have a common denominator, take the numerator of the dividend for a new numerator, and the numerator of the divisor for the new denominator. 2. When a fraction is to be divided by any quantity, it is the same thing whether the numerator be divided by it, or the denominator multiplied by it, 3. When

3. When the two numerators, or the two denominators, can be divided by some common quantity, let that be done, and the quotients used instead of the fractions first proposed,

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 result will be the power, the same as before.

Note. When the sign of the root is +, all the powers of it will be+; but when the sign is -, all the even powers will be +, and all the odd powers; as is evident from multiplication.

* 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 X a2 = a3 + 2 = a5. And a3a2 or