times 4 are 20, and 5, the tens of the 51, are 25, and 1 (written in the remainder,) are 26. We next consider the whole remainder, 177, as joined with 4, the next figure of the dividend, making 1774 for the next partial dividend. Proceeding as before, we obtain 313 for the second remainder, 217 for the final remainder, and 536 for the entire quotient. Hence, the following

RULE. I. Obtain the first figure in the quotient in the usual

manner.

II. Multiply the first figure of the divisor by this quotient figure, and write such a figure in the remainder as, added to this partial product, will give an amount having for its unit figure the first or right hand figure of the partial dividend used.

III. Carry the tens' figure of the amount to the product of the next figure of the divisor, and proceed as before, till the entire remainder is obtained.

IV. Conceive this remainder to be joined to the next figure of the dividend, for a new partial dividend, and proceed as with the former, till the work is finished.

GENERAL PRINCIPLES OF DIVISION.

113. The general principles of division most important in their application, relate; 1st, to changing the terms of division by addition or subtraction; 2d, to changing the terms of division by multiplication or division; 3d, to successive division.

114. The quotient in division depends upon the relative values of the dividend and divisor. Hence, any change in the value of either dividend or divisor must produce a change in the value of the quotient; though certain changes may be made in both dividend and divisor, at the same time, that will not affect the quotient.

CHANGING THE TERMS BY ADDITION OR SUBTRACTION.

115. Since the dividend corresponds to a product, of which the divisor and quotient are factors, we observe,

1st. If the divisor be increased by 1, the dividend must be increased by as many units as there are in the quotient, in order that the quotient may remain the same, (93, I); and if the dividend be not thus increased, the quotient will be diminished by as many units as the number of times the new divisor is contained in the quotient. Thus,

2d. If the divisor be diminished by 1, the dividend must be diminished by as many units as there are in the quotient, in order that the quotient may remain the same, (93, II); and if the dividend be not thus diminished, the quotient will be increased by as many units as the number of times the new divisor is contained in the quotient. Thus,

These principles may be stated as follows:

I. Adding 1 to the divisor takes as many units from the quotient as the new divisor is contained times in the quotient.

II. Subtracting 1 from the divisor adds as many units to the quotient as the new divisor is contained times in the quotient. Hence,

III. ADDING any number to the divisor SUBTRACTS as many units from the quotient as the new divisor is contained times in the product of the quotient by the number added; and SUBTRACTING

any number from the divisor ADDS as many units to the quotient as the new divisor is contained times in the product of the quotient by the number subtracted.

CHANGING THE TERMS BY MULTIPLICATION OR DIVISION.

116. There are six cases:

1st. If any divisor is contained in a given dividend a certain number of times, the same divisor will be contained in twice the dividend twice as many times; in three times the dividend, three times as many times; and so on. Hence,

Multiplying the dividend by any number, multiplies the quotient by the same number.

2d. If any divisor is contained in a given dividend a certain number of times, the same divisor will be contained in one half the dividend one half as many times; in one third the dividend, one third as many times; and so on. Hence,

Dividing the dividend by any number, divides the quotient by the same number.

3d. If a given divisor is contained in any dividend a certain number of times, twice the divisor will be contained in the same dividend one half as many times; three times the divisor, one third as many times; and so on. Hence,

Multiplying the divisor by any number, divides the quotient by the same number.

4th. If a given divisor is contained in any dividend a certain number of times, one half the divisor will be contained in the same dividend twice as many times; one third of the divisor, three times as many times; and so on. Hence,

Dividing the divisor by any number, multiplies the quotient by the same number.

5th. It a given divisor is contained in a given dividend a certain number of times, twice the divisor will be contained the same number of times in twice the dividend; three times the divisor will be contained the same number of times in three times the dividend; and so on. Hence,

Multiplying both dividend and divisor by the same number does not alter the quotient.

6th. If a given divisor is contained in a given dividend a certain number of times, one half the divisor will be contained the same number of times in one half the dividend; one third of the divisor will be contained the same number of times in one third of the dividend; and so on. Hence,

Dividing both dividend and divisor by the same number does not alter the quotient.

NOTE. If a number be multiplied and the product divided by the same number, the quotient will be equal to the number multiplied; hence the 5th case may be regarded as a direct consequence of the 1st and 3d; and the 6th, as the direct consequence of the 2d and 4th.

To illustrate these cases, take 24 for a dividend and 6 for a divisor; then the quotient will be 4, and the several changes may be represented in their order as follows:

which may now be stated as follows:

PRIN. I. Multiplying the dividend multiplies the quotient; and dividing the dividend divides the quotient.

PRIN. II. Multiplying the divisor divides the quotient; and dividing the divisor multiplies the quotient.

PRIN. III. Multiplying or dividing both dividend and divisor by the same number, does not alter the quotient.

118. These three principles may be embraced in one

GENERAL LAW.

A change in the dividend produces a LIKE change in the quotient; but a change in the divisor produces an OPPOSITE change in the quotient.

SUCCESSIVE DIVISION.

119. Successive Division is the process of dividing one number by another, and the resulting quotient by a second divisor, and so on.

Successive division is the reverse of continued multiplication. Hence,

I. If a given number be divided by several numbers in successive division, the result will be the same as if the given number were divided by the product of the several divisors, (95, I).

II. The result of successive division is the same, in whatever order the divisors are taken, (95, II).

CONTRACTIONS IN DIVISION.

CASE I.

120. When the divisor is a composite number.

1. Divide 1242 by 54.

OPERATION.

6) 1242
9) 207

ANALYSIS. The component factors of 54 are 6 and 9. We divide 1242 by 6, and the resulting quotient by 9, and obtain for the final result, 23, which must be the same as the quotient of 1242 divided by 6 times 9, or 54, (119, I). We might have obtained the same result by dividing first by 9, and then by 6, (119, II). Hence the following

RULE.

23 Ans.

Divide the dividend by one of the factors, and the quo