The new divisor need not be written each time as in the last example; it is sufficient to separate, in each successive division, a figure at the right in the first di

visor.

70. If the remainder of the first division is smaller than the divisor after its last figure is suppressed, a cipher must be put in the quotient; and if it is still small-, er, after another of the remaining figures is suppressed" in the divisor, a cipher must again be put in the quotient ;, and so on.

71. If, after commencing the operation by suppressing at the right of the dividend as many figures as the rule requires, the remaining figures do not contain the divisor; we must continue to suppress figures on the right of the divisor till it can be contained in the dividend.

72. The operation will be performed with greater exactness, if, after suppressing a figure in the divisor, the last of the remaining figures is augmented by unity, when the figure suppressed is equal to or greater than 5. In like manner the last of the figures remaining in the dividend should be augmented by unity, when the figures, suppressed agreeably to the direction of the rule, surpass 5,50, or 500, according as there are 1,2,3, &c.

73. From the preceding observations it is easy to learn how the quotient may be obtained to a much greater exactness. If, for example, the quotient is demanded to ten thousandths of an unit, or to the fourth place of decimals; the operation requires as many ciphers to be put after the dividend as there are decimals required in the quotient, Division is then performed according to the preceding method. And when the quotient shall have been found as for whole numbers, as many figures are separated at the right, as there are decimals desired.

EXAMPLE.

If the quotient of 6927 divided by 4532, is required to the fourth place of decimals, four ciphers are put after 6927; and the question is to find the quotient, in whole numbers, of 69270000 divided by 4552, that is, according to the preceding rule, to divide 69270 by 4532, as follows.

The quotient sought is 1,5285, to the fourth place of decimals.

If there are decimals in the dividend, in the divisor, or in both; they may be reduced to the condition of whole numbers by following the direction given in article (68;) the operation is then performed as in the last example.

9678

By observing the direction in article (71,) any given fraction may be readily reduced by this method to decimals. For example, if it be required to reduce 423 to. decimals, and to have the value to the third place of decimals; 4253000 must be divided by 9678; which (69) is performed by dividing 255 by 9678, or (71) by dividing 4253 by 968, according to the usual method. The quotient thus found is 439; hence, 0,439 is the value, to the third place of decimals, of $23.

74. It may nevertheless happen, that the quotient found by these rules will not be exact by 1, 2, or 3, units in the last figure. And although this case rarely occurs, it may not be improper to observe, that error may be prevented by separating at the commencement of the operation, on the right of the dividend, only two figures less than there are in the divisor, and then proceeding as before. When the quotient is found, the last figure in it is suppressed; and to the last remaining figure an unit is added, when the suppressed figure is greater than 5.

EXAMPLES FOR PRACTICE.

1. Reduce to a decimal whose value shall be found in three figures. Ans. ,375. 2. Required the value of 2 to two places of deci mals. Ans. ,88.

3. Required the value of 1 to the fifth place of de

cimals

4. Reduce to decimals.

Ans. ,00689.

Ans. 75..

PROOF OF MULTIPLICATION AND DIVISION.

75. The means of proving these two operations may be derived from the definition which has been given of them.

In multiplication, the multiplicand is taken as many times as there are units in the multiplier. Hence, if we seek how many times the product contains the multiplicand, that is, (59) if we divide the product by the multiplicand, the quotient will be the multiplier. And as the multiplier and multiplicand may be taken reciprocally for each other, if the product of a multiplication be divided by one of its factors, the quotient will be the other factor.

For example, having found in article (50,) that 2864 multiplied by 6, gives 17184 for product, we divide 17184 by 2864 and the quotient is the multiplier, 6.

In like manner, as the quotient in division expresses how many times the dividend contains the divisor; if the divisor be taken as many times as there are units in the quotient, that is, if the divisor be multiplied by the quotient, the product will be the dividend, if the division. was performed without a remainder. But if there is a remainder it must be added to the product..

For example, having found as in article (63,) that 189192 divided by 375, gives 505 for quotient and 117 for remainder; if we multiply 375 by 505 and add 117 to the product, the dividend, 189492, will be reproduced.

Thus Multiplication and Division reciprocally prove each other..

But these operations may be verified by a more ready method, which will now be explained. The preceding reflections, however, should not be neglected, as they will be useful on many occasions..

PROOF BY 9%

76. Let us suppose, that having multiplied 65498 by 454 and found 29736092 for the product, we wish to know if this product is exact..

The figures of the multiplicand 6, 5, 4, 9, 8, are add

ed together in the same manner as simple units, and their sum divided by 9. After division, 5 is found for emainder.

In like manner, the figures 4, 5, 4, of the multiplier, are added together, and their sum divided by 9. After division, 4 is found for remainder.

The remainders, 5 of the multiplicand, and 4 of the multiplier, are then multiplied together, and the product 20 divided by 9. After this division, 2 is found for remainder.

If the product to be verified is exact, on adding its figures 2, 9, 7, 3, 6, 0, 9, 2, and dividing their sum by 9, 2 will be found for remainder; which is the case.

As Division is an abridged method of performing Subtraction, this rule may be considered to be founded on this principle, that, to have the remainder after subtracting all the nines contained in a given number, we need only seek the remainder after suppressing all the nines contained in The sum of its figures added as simple units.

Thus, if all the nines are subtracted from a number expressed by a single figure followed by many ciphers, the remainder will be expressed by the same figure. If from 400, or 5000, or 60000, &c. all the nines are subtracted, the remainder will be 4, or 5, or 6, &c.

Then the remainder, which results from suppressing all the nines in such a number as 65498 (which is the same number as 60000, plus 5000, plus 400, plus 90, plus 8), will be the same as that given by 6, plus 5, plus 4, plus 9, plus 8; that is, the same as that given by adding all these figures as simple units.

Let us now explain the application of this proof to multiplication.

Since 65498 is composed of a certain number of nines and the remainder 5, and as the multiplier 454 is also composed of a certain number of nines and the remainder 4; in order that the total product may be divisible by 9, the product of 4 by 5, or 20 only is requisite; or, in casting out the nines, only 2 is wanting to render the total product divisible by 9.. Then there should be the same quantity remaining from the product, as in the product of the two remainders after casting out the nines they con-tain

The same proof may also be made by the figure 3. Division may be easily proved by observing what has been said in article (70.) After subtracting from the dividend the remainder given by division, the result may be considered as a product, of which the divisor and quotient are the factors; consequently, proof by 9 may be applied in the same manner as in the preceding case.

USES OF THE PRECEDING RULE.

77. Division is not only useful for finding how often onenumber contains another, but also for separating a number into equal parts. The half, third, fourth, fifth, twentieth, thirtieth, &c. part of a number may be taken by di-, viding the number by 2, 3, 4, 5, 20, 30, &c. or separ fing the number into 2, 3, 4, 5, 20, 30, &c. equal parts, to take one of those parts.

Division is also used for changing units of a certain kind into those of a higher denomination retaining the same value; for example, to reduce pence to shillings,. and shillings to pounds. It may be remarked, that when pence are to be converted into shillings, the pence must be divided by the number which expresses the number of pence in a shilling, and the quotient will be the shillings in the given pence. Thus, to reduce 5864 pence to shillings, we divide 5864 by 12, and 488, the quotient, is the number of shillings in 5864 pence. The remainder 8. is pence. To reduce 488 shillings to pounds, we divide 488 by 20, because 20 shillings make 1 pound, and the quotient 24 is pounds; and the remainder 8 is shillings.. So the total value of 5864 pence, expressed in pounds,. shillings, and pence, is £24 8s. Ed..

The principle explained in the preceding example may be summed up in this general rule. Divide the lower denomination by the number which expresses an unit of the next higher, and the quotient will be of the higher denomination. Divide this quotient by the number express-. ing an unit of the next higher denomination; and thus. continue to the last.. The last quotient and the several remainders will be equivalent to the number to be reduced.