CASE IV.

To find the Greatest Common Measure of the Terms of a Fraction.

DIVIDE the greater term by the less, and the last divisor by the last remainder, and so on till nothing remains; then the divisor last used will be the common measure required; just the same as in common numbers.

But note, that it is proper to range the quantities according to the dimensions of some letters, as is shown in division. And note also, that all the letters or figures which are common to each term of the divisors, must be thrown out of them, or must divide them, before they are used in the operation.

Therefore the greatest common measure is a + b.

INFINITE SERIES.

AN Infinite Series is formed either from division, dividing by a compound divisor, or by extracting the root of a compound surd quantity; and is such as, being continued, would run on infinitely, in the manner of a continued decimal fraction.

But, by obtaining a few of the first terms, the law of the progression will be manifest; so that the series may thence be continued, without actually performing the whole operation.

PROBLEM I.

To Reduce Fractional Quantities into Infinite Series by Division. DIVIDE the numerator by the denominator, as in common division; then the operation, continued as far as may be thought necessary, will give the infinite series required.

To Reduce a Compound Surd into an Infinite Series.

EXTRACT the root as in common arithmetic; then the operation, continued as far as may be thought necessary, will give the series required. But this method is chiefly of use in extracting the square root, the operation being too tedious for the higher powers.

EXAMPLES.

*To Subtract one Fractional Quantity from another. REDUCE the fractions to a common denominator, as in addi tion, 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.