DIVISION.

DIVISION is an operation by which we discover how often one given number is contained in another.

The number to be divided is called the Dividend, the number by which you divide is the Divisor, and that which expresses how often the Divisor is contained in the Dividend, is called the Quotient. If it happen that after the Divisor is taken as often as it can out of the Dividend, there be still something over, this is called the Remainder.

Of whatever denomination be the Dividend and Divisor, the Quotient is either of the same denomination, or an abstract number; as an expression of the magnitude of the former sum relative to the latter: but the Remainder is always of the same denomination with the Dividend. Thus if we divide £48. amongst 6 men, the Quotient 8 will represent the number of Pounds due to each: but if we wish to know how many yards of silk may be had for £. 1 when 36 yards cost £. 12, we divide the 36 yards by the 12 pounds, and the Quotient 3 shows the number of yards required.

As Multiplication is a short method of performing Addition, so Division is an abridgement of Subtraction: for instance, to know how often 6 is contained in 24, perform the repeated subtractions as here shown:

as in this example, we find that 4 times 6, will be just 24; we therefore say that 6 is contained 4 times in 24, and nothing remains over. By this method the operation is much shorter and more expeditious than by repeated subtractions.

To assist in this operation you may make use of the Multiplication Table already given, in this manner: run along the first line for the figure 6, and going down the column under 6, you will find 24, on a line at the beginning of which stands the figure 4, which will be the quotient. Again, to divide 49 by 8; in the column under 8, find the number nearest to, but less than, 49 which will be 48, and going to the beginning of the line on which you see 48, you will find 6 for the quotient: and the difference between 48 and 49, that is 1, will be the remainder or in other words, 8 will be contained 6 times in 49, and there will be 1 over.

In this manner when the Divisor is 12 or under, the Multiplication Table will be serviceable; but when it exceeds 12, we try any Multiplier that promises to answer; and if the product is greater than the Dividend, the Multiplier or Quotient is too great; on the contrary if the product falls short of the Dividend by a sum greater than the Divisor, or equal to it, the Multiplier is too small; and by repeated trials, at last the proper Multiplier or Quotient is discovered.

first 68, and ask how often 8 may be had in 68. By the Multiplication Table you will find that 8 times 8 are 64,

and

and 9 times 8 are 72, which last being more than the given sum 68, you must take the product below it, that is 8 times 8 or 64. Then write this 8 on the right-hand of the dividend, beyond the separating line, placing the product 64 under 68, drawing a line under these figures, in order to subtract the less from the greater, which will give a remainder of 4. Hence it appears that 8 may be contained 8 times in 68 and 4 will remain. To this 4 bring down the next figure of the dividend, which is 7, placing a dot under it, to prevent mistakes, by showing that it has already been used.

Now we have a new dividend 47, in which 8 will be contained 5 times: this 5 is written in the quotient, fol lowing the 8, and multiplying the divisor 8 by 5 we have 40, to be written under 47, and subtracting the less from the greater, the remainder will be 7; to this 7 bring down the next figure of the dividend, 3, making 73, and, as before, ask how often 8 can be taken out of 73. By the Table it will appear that 9 times 8 are 72, the nearest number under 73; and writing the 72, subtract it from 73, which will give 1 for the remainder. All the figures of the dividend being now exhausted, the division is performed, and we find that 8 is contained 859 times in 6873, and that there is 1 remaining as a surplus.

Again in working with a divisor of more than one place of figures, as in the following example, to divide 249295 by 365, write down the divisor and dividend as in the last example, thus,

and counting how many places of figures are in the divisor, take the same number in the dividend from the left hand, and say how often can I take 365 out of 249? As this cannot be done, take in another place of figures and say how often can I have 365 out of 2492? If the divisor consisted only of the figure 3, we should have it 8 times in the 24 of the dividend: but 36 could not be taken 8 times out of 249; and even if it were tried to take 365 seven times out of 2492, it would be found impossible: let 6 then be tried, and multiplying 365 by 6, place the product 2190 under the dividend, and drawing a line, subtract this sum from the figures above, by which a remainder will be found, 302. To this sum bring down the figure 9, which is next after those already employed, and say how often can the divisor 365 be found in 3029: This upon trial will be found to be 8 times, which multiplied into 365 will give 2920; and this subtracted from the former remainder with its additional figure 9, will leave another remainder 109. To this bring down the last figure of the dividend 5, and say how often can the divisor 365 be obtained out of 1095; This will be found to be 3 times; and multiplying 365 by 3, you will have 1095, equal to the sum above it, and leaving therefore no remainder.

A number which divides another into any parts, without leaving a remainder, is called a Measure of that number: thus 2, 4, 5, and 10, are measures of 20, because each of them will exactly divide 20, without any remainder : and these parts or measures are also termed aliquot parts: hence 1 penny, 2d. 3d. 4d. 6d. are aliquot parts of a shilling, &c.

To assist in discovering the measures of any given numbers, it must be remembered that numbers ending with an even figure, such as 2, 4, 6, 8, or 0, are all measurable by 2-That every number ending with 5 or 0, may be measured by 5-That all numbers, whose figures, when added

together

together, give an even number of threes or nines, may be measured by 3 or 9 respectively.

When it happens, as in the following example, that the remainder together with the figure brought down from the dividend, is not equal to the divisor, you must write 0 in the quotient, and bring down another figure, to be proceeded with as before. This must be done, and 0 placed in the quotient, as often as the sum to be divided is less than the divisor.

Here the number to be subtracted from the second dividend, leaves only a remainder of 16, to which 1 being brought down, makes the new dividend 161 which being less than the divisor 5386, it cannot be divided by that divisor; write therefore 0 in the quotient and bring down another figure 5. The sum to be now divided, 1615, being still less than the divisor, cannot be divided by it; you will therefore write another 0 in the quotient, and bring down the last figure of the dividend 8, making a sum of 16158, which is not only greater than 5386, but will contain it 3 times and leave no remainder,

Division may be contracted in various ways; as for instance, in dividing by any of the digits or 10, 11, & 12, write down the divisor and dividend as before?

3 124

Divr. 5)8536295(0 Remr.

Quot. 1707259

9)25603976(2 Remr.

2844886. Quot.

and drawing a line under the dividend, say, 5 in 8 once and

2 C 2

8 over: