Q. Why is division called a compendious method of subtraction?

A. That division is a short method of subtraction I prove thus: Suppose 24 dollars were to be divided between 6 men equally

24

First, I would give each man one, viz.

Now, as I have given each man one 4 times, each man must have 4 dollars; but this being too tedious, I take a quicker method, and try how often 6 is contained in 24.

6)24(4 times, or 4 dollars to each man.

RULE.-Set the divisor to the left hand of the dividend, and try how often the divisor is contained in the highest figure of the dividend; but if the said figure be less than the divisor, take the two first figures, and try how often the divisor is contained therein, noting the multiplying figure for the quotient; the product of which subtract from the said figures; to the remainder bring down the next figure, which put in unit's place: this number becomes the dividend. Then proceed as before, till you have all the figures in the dividend brought down. The numbers thus found are called the quotient.

Q. Suppose the divisor contains more figures than one?

A. Compare the highest figure in the divisor with the highest figure in the dividend, and see how often it is contained, making an allowance for what may be carried from the foregoing figure. Or, if the highest figure in the dividend is less than the highest in the divisor, compare the highest figure in the divisor with the two highest in the dividend, making an allowance, as above. When this is found, proceed, as before taught, by multiplying the divisor from unit's place, putting the multiplying figure as first or highest figure in the quotient, and the product under the leading figures in the dividend. Subtract, and proceed as before taught. 856543)4276546746(4992

376500004)800000000000

60000004)60000000000000

To prove division, multiply the quotient and divisor together,. adding in the remainder. If the product agrees with the dividend, the work is right.

The most expeditious method is by casting the nines out of the different numbers, as follows :—

Make a X. Cast the nines out of the divisor-set the excess on the left. Cast the nines from the quotient-set the excess on the

right. Multiply these figures together, cast the nines from the product, and carry the excess to the remainder; from which cast the nines, and set the excess at the top. Then cast the nines from the dividend, and set the excess at the bottom. If the top and bottom figures agree, the work is right.

Short Division.

To divide by 12, or any figure under, in a line, try how often the divisor is contained in the first part of the dividend as before, making the subtraction in your mind, and carry the difference to the next figure, setting no figure down but in the quotient.

EXAMPLE.

6)45643854

7607309

Here I try how often 6 is contained in 45. I find 7 times, which I place under the second figure in that part of the dividend, and carry 3 (the remainder) to 6—36, which, divided as before, quotes 6, no remainder. Then 6 in 4 nought times; but take 4 to 343, divided by 67 times, &c.

If a divisor be such, that the product of any two figures will measure it, divide the dividend by either of these figures, and the quotient found by the other figure. This last quotient will be the

Here I divide mentally by 7, and find 4 the remainder; then I divide by 5, the next figure, and find my quotient with 2 remainder. Then multiply the first divisor by this last remainder, and add in the first remainder, which equals 18, the full remainder at the end of the division.

In dividing by 10, 100, 1000, &c. cut off as many figures from the right of the dividend as the divisor contains units, and the work is done. If I divide 47634 by 100, I point off 34 for the two cyphers in 100, which 34 is the remainder.

Problems resulting from the foregoing Rules.

1. Having the sum of two numbers, and one of them given to find the other; subtract the given number from the given sum. The remainder is the number required.

Let 475 be the sum and 144 the number known. Required the other.

2. Having the greater of two numbers, and the difference of it and a less, given to find the less. Subtract one from the other.

3. Having the less, and difference given to find the greater. Add them together.

475 Sum.

144 Given number.

331 Required.

475 Greater.
331 Difference.

144 Less required.

144 Less.
331 Difference.

475 Greater.

4. Having the product of two numbers, and one of them given to find the other. Divide the product by the given number, the quotient will be the number required.

Let the product of two numbers be 120, and 8 the number given, required the

other.

8)120

15 Number sought.

5. Having the dividend and quotient to find the divisor. Divide the dividend by the quotient.

COROLLARY.-Hence we find another method of proving di

vision.

6. Having the divisor and quotient given to find the dividend. Multiply them together.

Questions to exercise the foregoing Rules.

By an application of the foregoing problems, the following questions may be elegantly solved.

1. What is the difference, and what is the sum of six dozen dozen and half a dozen dozen? Ans. 936 sum, 792 difference.

2. The remainder of a division sum is 432, the quotient is 423, the divisor is the sum of both and 19 more: What is the dividend? Ans. 200934.

3. There is a certain number which, being divided by 7, the quotient multiplied by 3, that product divided by 5, from the quotient subtract 20, to the remainder add 30, and half the last sum shall be 35.

Ans. 700.

4. What number is that which, being added to 9709, will make 10901? Ans. 1192.

5. A sheep-fold was robbed three nights successively, the first night half the sheep were stolen, and half a sheep more; the second night half the remainder were taken, and half a sheep more; the third night they took half of what were left, and half a sheep more; by which time they were reduced to 20: How many were there at first ?. Ans. 167. 6. What number must I multiply by 7, that the product may be 623? Ans. 89. 7. The product of two numbers is 31383450, and one of them 4050: The other factor is required. Ans. 7749.

8. What number deducted from the 26th part of 2262 will leave the 87th part of the same?

Ans. 61. 9. There are two numbers, the greater is 73 times 109, and their difference is 17 times 28: What is their sum and product? Ans. Sum 15438. Product 59526317.

Numbers of divers Denominations.

NUMBERS of divers denominations are those whereby we express the sundry divisions of Money, Weight, and Measure.

The following weights and measures, as established in the U. S. of America, are agreeable to the standard weights and measures preserved in the Exchequer of Great Britain, under the care of

By Troy weight is weighed gold, silver, jewels, and liquors.

The standard for the gold coin of the United States of America is 11 parts of pure gold melted with one of alloy, which is of the same fineness of British gold.

The standard for the silver coin of the United States is 1485 parts of pure silver, and 179 parts of alloy, which must be wholly of copper.