DIVISION.

TO know how many times one number is contained in another, is the use of Division. It teaches also, to separate any number, or quantity, into any number of parts assigned; and shows how, from any two numbers given, you may find a third, which shall consist of so many units, as the one of those given numbers is contained in the other.

In Division there are four principal parts to be understood. The dividend, or number to be divided; the divisor, or number by which you divide; the quotient, or answer to the question, which shows how many times the divisor is contained in the dividend, and the remainder, which is always less than the divisor; and of the same denomination with the dividend. The remainder is uncertain: for there is sometimes a remainder, and sometimes none.

Division is either simple or compound. Simple, when the divisor consists of but one figure; and the dividend, of two, or more. Compound, when the divisor consists of two, or more, figures.

TO PROVE Division, you may multiply together the quotient and divisor, taking heed to add the remainder, if any there be. If the product be like the dividend, your work is right; if otherwise, it is wrong.

CASE I. and RULE.

Inquire, first, how many times you can have your divisor in the first figure of the dividend.* When known, place it in the quotient; then multiply the divisor by this quotient figure, and set the product under the left hand figure, or figures, of the dividend, as the case may be; then subtract this product from the figure or figures, of the dividend, under which it is placed, and bring down the next figure of the dividend to the right hand of the remainder; then proceed as in the first instance. If the figure, brought down, be less than the divisor, you must place a cypher in the quotient, and bring down another figure to make a second dividend. You must proceed carefully, in this manner, wich all the figures of the dividend, till your work is finished.

EXAMPLES. 7)365(52

35

15

14

(1)

* If you cannot have your divisor in the first figure of the dividend, you must take two, three, or four figures, as the case may require.

4)9236(
6)7436(

3)6969(

8)74236(

4)862468(

9)723642(

Multiplying the quotient by the divisor, is a sure way of prov. ing division, as already mentioned. But long division, may be proved by addition.

1 Proof by Addition.

RULE.

Add together the remainder and all the bottom lines, and if their sum be like the dividend, the work is right.

1. 23)44(1

2.

26)742(28

3.

37)8236(222

N. B. The asterisms show the bottom lines and the remainder, which are to be added together, as proof.

Examples, in which only the remainder, and the proof by the excess of nines, are set down.

4)63426742(6)72314267(5)4236742( 3)14674236(

14)6234674( 18)67423674( 28)62347742( 54)62342674(

CASE 2.

When the divisor does not exceed 12, the operation, is called short division.

RULE.

Inquire how often you can have the divisor, in the first figure, or figures, of the dividend. Then multiply, in your mind, the divisor by the quotient figure, and subtract the product from a like number of figures, at the left hand of your dividend. The unit, or units, which remain, if there be any, must be reckoned as so many tens, which you must consider as standing at the left hand of the next figure of the dividend, and to be reckoned with it; then inquire how often you can have your divisor in these two figures. If nothing remain, you must inquire how often you can have the divisor in the next figure, and thus proceed, till the work is done.

In dividing by 110, 120, 1100, or 12000, &c. the learner has nothing to do, but to cut off, or separate the cyphers, in the divisor, 110, 120, &c. and cutting off, or separating a like number of figures from the right hand of the dividend.

7)842364936423

CASE 3.

12000)2367426378 19000)634267894 19000)7236423655

11)72646206 12)76677240 11)47627000

12)42007400

By fully understanding the above examples, you may expedi tiously divide, by 110, 120, 1100, or 1200, &c. For, in the operation you have nothing to do, except cutting off, or separating the cyphers from 11, and 12, (when these numbers happen to be the divisors) and separating, or cutting off, the like num. ber of figures, or cyphers, from the right hand of your dividend. And then proceed, as in the above examples.

Divisor 11)0)3456(7

314

CASE 4.

To divide by 10, 100, 1000, 10000, 100000, &c.

RULE.

Cut off, or separate, as many figures or cyphers, from the right hand of your dividend as you have cyphers in the divisor, and your work is done; for the remaining figures of your dividend will be the quotient, and those cut off, the remainder.

To exercise the pupil, we shall add some promiscuous exam. ples and questions, under addition, subtraction, multiplication, and division,

1. Add together 42, 602, 7046, 47823, 460786, and 74. Ans. 516373.

2. What is the number, being added to 24978792879, that will produce 46324674236? Ans. 21345881357.

3. John owes Peter 6342 dollars, and has paid 5986, what sum is still due to Peter? Ans. 356.

4. The amount due from A. B. C. and D. to F. is 63427 dollars A. has paid 279; B. 3784; C. 742, and D. 46. What is still due to F? Ans. 58576 dols.

5. Six men in partnership, have, in stock, 74628 dollars, of which M. put in 436; L. 792; S. 4623; N. 6742; and Q 2763; what did H. put in ? Ans. 59272,

6. How many pence are there in one dollar, a half dollar, quarter of a dollar, a shilling, and six pence, estimating a dol. at 8 shillings? Ans. 186 pence.

7. A vessel, containing 422 pieces of nankeen; 456 chests of Hyson tea; 397 pieces muzlin; 4276 yards silk; 674 yards cassimer, and 7496 yards of chintzes, was taken by 86 men, and equally divided among them. How much of each kind fell to each man's share? Ars. 4 pieces nankeen, and 78 remaining: 5 chests of Hyson, and 26 remaining; 4 pieces muzlin, and 53 remaining; 49 yards of silk, and 62 remaining; 7 yards cassimer, and 72 remaining; 87 yards chintzes, and 14 remaining.

8. The undivided remainders of the above cargo were sold for 5276 dollars. How many dollars fell to each man's share? Ans. 61 dollars, 34 cents, 8 mills, and 72 of a mill.

9. Socrates, the famous Grecian philosopher, was put to death 400 years before the birth of Christ.-General Washington died 1799 after the birth of Christ. What is the difference of time? And how old would Socrates be, if he had lived to this year 1806? 1 Ans. 2199, 2 Ans. 2206. The last answer makes no allowance for Socrates' age.

10. The world was created 4004 years before the birth of Christ. Gen. Washington was born 1732 years after the birth of Christ. How old was the world when Washington was born! Ans. 5736.

COMPOUND ADDITION.

COMPOUND Addition is the adding of several numbers together, having divers denominations.

RULE.

1. Place the numbers of a similar denomination under each other, and separate each denomination, by a comma. The lowest denomination should, ever, be in the right hand column. 2. Begin with the right hand column first; add it up, and see how many of the next denomination are contained in the first column, carry the ones, or the sum to the second column, set the overplus directly under the first column. Then begin with the second column, and proceed in the above manner, till the operation be finished.