CASE I.

When the Divisor and Dividend are both Simple Quantities;

SET the terms both down as in division of numbers, either the divisor before the dividend, or below it, like the denominator of a fraction. The abbreviate these terms as much as can be done, by cancelling or striking out all the letters that are common to them both, and also dividing the one co-efficient by the other, or abbreviating them after the manner of a fraction, by dividing them by their common

measure.

Note. Like signs in the two factors make + in the quetient; and unlike signs make; the same as in multiplication*.

Because the divisor multiplied by the quotient, must produce the dividend. Therefore,

1. When both the terms are +, the quotient must be +; because + in the divisor in the quotient, produces + in the dividend. 2. When the terms are both, the quotient is also + because ➡in the divisor × + in the quotient, produces—in the dividend.

3. When one term is and the other, the quotient must be -: because in the divisor X - in the quotient produces in the dividend, or in the divisor x in the quotient gives in the dividend. So that the rule is general; viz. that like signs give +, and unlike signs give, in the quotient.

VOL. I.

25

CASE

CASE II.

When the Dividend is a Compound Quantity, and the Divisor a Simple one:

When the Divisor and Dividend are both Compound Quantities:

1. SET them down as in common division of numbers, the divisor before the dividend, with a small curved line between them, and ranging the terms according to the powers of some one of the letters in both, the higher powers before the lower.

2. Divide the first term of the dividend by the first term of the divisor, as in the first case, and set the result in the quotient.

3. Multiply the whole divisor by the term thus found, and subtract the result from the dividend.

4. To this remainder bring down as many terms of the divided as are requisite for the next operation, dividing as before; and so on to the end, as in common arithmetic.

Note.

CASE L

When the Divisor and Dividend are both Simple Quantities;

SET the terms both down as in division of numbers, either the divisor before the dividend, or below it, like the denominator of a fraction. The abbreviate these terms as much as can be done, by cancelling or striking out all the letters that are common to them both, and also dividing the one co-efficient by the other, or abbreviating them after the manner of a fraction, by dividing them by their common

measure.

Note. Like signs in the two factors make + in the quetient; and unlike signs make -; the same as in multiplication*.

Because the divisor multiplied by the quotient, must produce the dividend. Therefore,

:

1. When both the terms are +, the quotient must be +; because + in the divisor in the quotient, produces + in the dividend. 2. When the terms are both, the quotient is also +; because in the divisor X + in the quotient, produces - in the dividend. 3. When one term is and the other, the quotient must be because in the divisor X - in the quotient produces in the dividend, or in the divisor x in the quotient gives — in the dividend. So that the rule is general; viz. that like signs give +, and unlike signs give, in the quotient.

CASE

CASE II.

When the Dividend is a Compound Quantity, and the Divisor a Simple one:

4. Divide 6ab

5. Divide 3x2

30az 48z

8axa by 2a.

=30a

48.

15 + 6x + 6a by 3x.

6. Divide 6abc + 12abx 9a2b by 3ab.

7. Divide 10a2x 15x2

--

25x by 5x.

5ac.

8. Divide 15a2bc 15acx2 + bad by 9. Divide 15a + 3ay 18y* by 21a. 10. Divide 20ab60ab3 by

CASE III.

6ab.

When the Divisor and Dividend are both Compound Quantities :

1. SET them down as in common division of numbers, the divisor before the dividend, with a small curved line between them, and ranging the terms according to the powers of some one of the letters in both, the higher powers before the lower.

2. Divide the first term of the dividend by the first term of the divisor, as in the first case, and set the result in the quotient.

3. Multiply the whole divisor by the term thus found, and subtract the result from the dividend.

4. To this remainder bring down as many terms of the dividend as are requisite for the next operation, dividing as before; and so on to the end, as in common arithmetic.

Note.

CASE I.

To reduce a Mixed Quantity to an Improper Fraction.

MULTIPLY the integer by the denominator of the fraction, and to the product add the numerator, or connect it with its proper sign, or; then the denominator being set under this sum, will give the improper fraction required.

EXAMPLES.

a x x b

19

5

the Answer.

the Answer.

x

and a

to improper fractions.

CASE IL

To Reduce an Improper Fraction to a whole or Mixed Quantity. DIVIDE the numerator by the denominator, for the integral part; and set the remainder, if any, over the denominator, for the fractional part; the two joined together will be the mixed quantity required.

EXAMPLES.