Illustration of the process. 3 times 8 are 24, I write 4, and carry 2; 3 times 6 are 18, and 2, which I carried, are 20, I write 0, and carry 2; 3 times 4 are 12, and 2, which I carried, are 14, I write 4, and carry 1; 3 times 5 are 15, and 1, which I carried, are 16, which I write.

Illustration of the proof. In the multiplicand, 5 and 4 are 9; 6 and 8 are 14, or 5 over 9, which I write on the left of the cross. I write the multiplier 3 on the right. Now 5 on the left multiplied by 3 on the right, make 15, or 6 over 9, which I write at the head of the cross. In the product, 1 and 6 are 7 and 4 are 11, or 2 over 9; 2 and 4 are 6, which I write at the bottom of the cross. As this agrees with the 6 at the top, it affords a strong presumption that the work is accurate.

The Rule for Multiplication is founded on the following axioms:

1. The product of any digit into a number of several places, is equal to the sum of the products arising from the multiplication of that digit into every place of the multiplicand: thus 24×5=4×5 + 20 × 5.

2. If the multiplier be divided into any number of parts, the product of the whole multiplier into the whole multiplicand, is equal to the sum of the products arising from the multiplication of every part of the multiplier into the whole multiplicand: thus, 24 × 54 = 24 × 4+ 24 × 50.

3. Whatever is the product of two digits, that product will be 10 times as great, if one of the factors be advanced a place higher; and 100 times as great, if both factors be advanced a place higher and, in general, every advance of place in either of the factors, increases the product 10 times.

EXERCISES.

RULE II.

When the multiplier is equal to the product of two or more numbers, none of which exceeds 12, multiply by these numbers su sively.

EXAMPLE.

Multiply 34175 by 25.

34175

DIVISION.

DIVISION is a short way of finding out how many times a less number can be subtracted from a greater; or, it is that process by which we discover how often one number, called the divisor, is contained in another number called the dividend; and the answer is called the quotient.

RULE.

Draw a curved line on each side of the dividend, and write the divisor on the left of it. Then point off from the left of the dividend as many figures as will contain the divisor less than ten times for a DIVIDUAL, (or partial dividend); try how often this divi dual contains the divisor, write the answer, for the first figure of the quotient, on the right of the dividend, multiply the divisor by this figure, subtract the product from the dividend, and to the right of the remainder annex the next figure of the dividend for a new dividual; from which find, as before, the next figure of the quotient'; and so on, till all the figures of the dividend have been used. If it be necessary to annex more than one figure of the dividend to any remainder to form a dividual that will contain the divisor, a cipher must be written in the quotient for every additional figure so brought down. The several quotient figures thus found, considered as one number, make the true quotient. When the work is done, if there be a remainder, annex it to the quotient, putting it above a line, and the divisor below, in the form of a fraction.

Note 1. When the divisor is 10, 100, 1000, &c. cut off from the right of the dividend, for a remainder, as many figures as there are ciphers in the divisors the figures which remain on the left are the quotient.

2. When there are ciphers on the right of the divisor, cut them off, together with an equal number of figures from the right of the dividend; then divide, as usual, by the remaining figures of the divisor, and when the work is done, annex the figures which were cut off from the dividend to the last remainder.

PROOF. Multiply the quotient by the divisor, to the product add the remainder; then will the sum, if the process has been accurately performed, be the same as the dividend.

The following is an easier proof: Cast out the 9s from the divisor and quotient, multiply the excesses together; from their pro. duct cast out the 9s, and add the excess to the remainder; then if the excess of 9s in the sum be not equal to the excess of 9s in the dividend, the work is inaccurate.

C.

Illustration of the process. The first dividual is 224, in which the divisor 52 is contained 4 times; I therefore write 4 as the first figure of the quotient, multiply 52 by 4, subtract the product 208 from 224, and to the remainder 16 annex 0, the next figure of the dividend, for a new dividual. In 160, the divisor 52 is contained 3 times; I therefore write 3 as the next figure of the quotient. multiply 52 by 3, subtract the product 156 from 160, and to 4, the remainder, annex 9, the next figure of the dividend. Then, because 52 is not contained in 49, I write 0 in the quotient, and to 49 I annex 1, the last figure of the dividend, for a new dividual. In 491, the divisor 52 is contained 9 times; I therefore write 9 in the quotient, multiply 52 by 9, subtract the product 468 from 491, and there remain 23. Thus the true quotient is 4309.

REMARKS.

1. When the divisor is large, to find how often it is contained in any dividual, consider how often its first figure is contained in the first, or first and second figures of the dividual, and you will obtain the quotient figure nearly, which diminish, till you find it

answer.

2. If any product exceed the dividual from which it is to be subtracted, the quotient figure is too great, and must be expunged, and a less figure used.

3. If any remainder be equal to, or greater than the divisor, the quotient figure is too small, and must be rubbed out, and a greater figure used.

4. To prevent mistakes, it is usual to make a dot under each figure of the dividend as it is brought down.

5. Division being just the reverse of Multiplication, its object may be effected, by successively subtracting the divisor from the dividend, as often as it can be done; then the number of subtractions will express the quotient. But this process, though simple, is too tedious for practice: it may, however, be easily abridged; for, if we multiply the divisor by any number, and subtract the

product from the dividend, the effect will evidently be the same, as if we subtracted the divisor from the dividend as many times as the number we multiply by expresses.

On this principle, if we find by trial, the greatest digit of the highest rank, by which when the divisor is multiplied, the product does not exceed the dividend, and subtract that product from the dividend, and then repeat the same process with the remainder, when there is one; and so on, till either nothing remains, or a remainder is obtained less than the given divisor; the sum of the values of the several multipliers so found, will evidently express the quotient, or the number of subtractions required to exhaust the di

vidend.

on,

Or, if we resolve the dividend into parts, and divide the largest part in the manner just now stated, and if there be a remainder, add it to the next part, and then divide the sum as before; and so till we have gone through all the parts: the sum of the quotients, (or the figures of the quotient placed so as to make a number equal to that sum,) will evidently be the quotient of the dividend divided by the divisor, because the divisor is thus taken out of every part as often as possible, and the whole is equal to all its parts taken together. The rule given for Division differs from this last process only in omitting the ciphers on the right, which are evidently unnecessary to be retained, as they do not affect the result. An example will illustrate this reasoning, and shew the founda tion of the rule.

Divide 8815 by 25.

Here we may consider 8815 as=8800+10+5.