repeated; it must ninish it to nothing. How is multiplication proved by division? A. By dividing the product by either of the factors; and if the quotient be equal to the other, the work is right; because division is only the reverse of multiplication, or a short way of performing subtraction; and if we divide the product by the multiplicand the quotient must equal the multiplier, which shows how often the multiplicand has been subtracted from the product; and as it can only be subtracted as often as the multiplicand Las been repeated, the quotient must always equal the multiplier, leaving no remainder. When the multiplier consists of more than one figure, how do you proceed? A. First, multiply the multiplicand by the right hand figure of the multiplier; then by the next left hand figure of the multiplier; and so on, till the multiplicand has been repeated by each figure of the multiplier, placing the product of each directly under the multiplying figure; then add the several products, in the same order in which they stand, for the total product. What must be done, when ciphers occur be tween the significant figures of the multiplier? A. Care must be taken to commence the product of the first figure of the multiplier, in each place, directly under the figure by which we multiply. If our multiplier is 36, and instead of removing the product of the 3 one figure further to the left than the product of the 6, we place the product of the 3 directly under the product of the 6, and then add the products, what would it be the same as multiplying by? A. By 9; because we first repeat our multiplicand 6 times, and then 3 times; and not giving the product of the 3 its proper local place, the multiplicand stands 6 times and 3 times repeated; that is, when added, 9 times repeated; but had we placed the product of the 3 one figure further to the left, we would have given it its local value; the products then would have expressed the multiplicand 6 times repeated, and 30 times repeated, and when added, the multiplicand 36 times repeated. When ciphers occur at the right hand or one or both the factors, what must be done? A. The significant figures must be multiplied the same as if no ciphers. were at the right; then to the right hand of the total product of the significant figures, we annex as many ciphers as are equal to the ciphers at the right of both the factors; in order to give the significant figures their local place in the product. How do you multiply by 10, 100, 1000, &c. ? A. To multiply by 10, we join a cipher to the right of the multiplicand, which increases it ten times; because figures increase in a tenfold proportion, and the cipher removes the place of each one figure further to the left; the figure which first expressed units now expresses tens, and the rest at the left increase in the same proportion; and when we join two ciphers, the unit figure becomes hundreds; and the figure in the place of hundreds expresses 100 times its simple value; and when we join three ciphers, the unit figure becomes thousands, that is, one thousand times its simple value; and the figures at the left, being in a like manner removed to the left, increase in the same proportion. Can you multiply by 9, 99, 999, &c. in any other way than by directly multiplying the multiplicand by the number of nines? A. Yes, we may join as many ciphers to the right of our multiplicand as our multiplier has nines, and from the sum thus produced, subtract the multiplicand; because by joining one

cipher to the multiplicand, we'repeat it 10 times; and then subtract ing the multiplicand once, leaves it 9 times repeated; and if our multiplier is 99 we join two ciphers which increases our multiplicand 100 times; then subtracting the multiplicand once, leaves it 99 times repeated; and we observe the same rule, in multiplying by 3 nines, or any number of nines. How do you proceed to multiply by a composite number? A. First multiply the multiplicand by one of the component parts, and then that product by the other; and the last product will be the total product. When is a number said to be a composite number? A. When it is produced by multiplying together any two figures in the multiplication table. Is thirty-five a composite number? A. Yes. What are its component parts? A. 7 and 5. Why? A. Because 7 times 5 are 35. In multiplying by the component parts of 35, why should it produce the same product, as multiplying directly by 35? A. Because when we multiply by 7, we repeat our multiplicand 7 times; and when we multiply this product by 5, we repeat a number 5 times, which is 7 times greater than our first multiplicand; and 5 times 7 times must make 35 times repeated.- What is multiplication of decimal, or federal money? A. It is repeating the parts of an integer, or integers and parts of integers, a given number of times.What are integers in federal money? A. Dollars. What are the de cimal parts of an integer? A. Dimes, cents and mills, or cents and mills, as they are commonly expressed. What do you understand by a decimal? A. I understand, that it is something less than a unit; in calling tenths, I understand that it takes 10 for a unit; and hundredths, that it takes 100 for a unit; and thousandths, that it takes 1000 to equal a unit; so that the name of each figure, shows the exact number required to equal a unit. How do you multiply decimal money? A. The same as whole numbers. How do you distinguish decimals from whole numbers? A. By a separatrix placed between them. Why are decimals multiplied the same as whole numbers? A. Because they increase in a tenfold proportion, the same as whole numbers. How do you point off decimals in your product? A. As far from the right, as the separatrix is in the multiplicand. When the price of one yard, one pound, or the price of a unit of any kind, is given; how do you obtain the price of a quantity? A. By multiplying the price of one, by the number expressing the quantity. Why should that give the price of the quantity? A. Because it is repeating the price of one as many times, as the quantity expresses, or contains units. If one yard cost $4,50 cts.; how would you find the price of 4 yards? A. I would multiply $4,50 cts., the price of one yard, by the units expressed in the number of yards, that is, by 4. Why should that give the price of 4 yards? A. Because, it is plain, that 4 yards must cost 4 times as much, as one yard; and multiplying the price of one yard by 4, is repeating the price of one yard, 4 times.

SIMPLE DIVISION,

Is the shortest method of performing subtraction, where the same number is to be taken away a given number of times. It shows in a

concise manner how often one number is contained in another, or how often one number may be subtracted from another. The two given numbers in division are called divisor and dividend. The number arising from the operation of the work, is called the quotient. And there is sometimes an uncertain number called the remainder, The dividend is the number to be divided. The divisor is the number by which the dividend is to be divided. The quotient shows how many. times the dividend contains the divisor, or how many times the divi sor can be subtracted from the dividend. The remainder is something left of the dividend after the divisor can no longer be subtracted, and is always less than the divisor.

In multiplication we have the price of a unit given to obtain the price of a quantity. In division we have the price of a quantity given to obtain the price of a unit; so it will be perceived, that division is exactly the reverse of multiplication, as well as a short way of performing subtraction; by it, we change lower denominations into higher; and by it we divide any number or quantity into an equal number of parts or shares, containing equal quantities.

RULE. Place the divisor at the left hand of the dividend, separated from it by a vertical line, (I). Then, assume as many of the left hand figures of the dividend, as shall be sufficient to contain the divisor something less than ten times, and inquire how often the di-, visor is contained in the assumed number, and place the result for a quotient, at the right of the dividend, separated therefrom by a small vertical line. Multiply the divisor by this quotient figure, and place the product under the assumed part of the dividend. Then subtract it therefrom, and to the remainder, bring down the next figure of the dividend, which must be divided as before, placing the result in the quotient; and so proceed, till all the figures of the dividend, have been brought down. After multiplying by your quotient figure, if the product be greater than the assumed part of your dividend, you must lessen your quotient figure; but if the remainder (after subtracting) be greater than the divisor, you must increase your quotient figure.And when you bring down (to the right of your remainder) a figure from the dividend, and it does not then contain the divisor, you must write a cipher in the quotient, to signify, that the divisor is not contained, and then bring down another figure from the dividend; and so continue to do, till your divisor is contained, or all the figures brought down from the dividend.

METHODS OF PROOF.-The best method of proving division, is by multiplication.

1. Multiply the quotient by the divisor, and if the product be equal to the dividend, the work is right. But should you have a remainder, it must be added to the product to make it equal to the dividend.

2. Division may be proved by addition. Add the divisor as many times, as the quotient expresses a unit: and to the amount add the remainder, (if you have any,) and if the sum equal the dividend, the work is right. But the better way to prove division by addition, is to add the remainder to the several products produced from the multiplication of the divisor in the order in which they stand; and if the amount equal the dividend, the work is right.

3. Division may be proved by subtraction. Subtract the divisor from the dividend, as often as the quotient contains a unit; and if it diminish it to nothing, the work is right; but when you have a remainder in dividing, your remainder after subtracting, must equal your remainder, when the work was performed by division.

4. Division may be made to prove itself. Divide the dividend by the quotient, and if your quotient equal your divisor in the first operation, the work is right; and if you had a remainder in the first work, the remainder in the last operation must equal the remainder in the first.

Division is denoted by the following character; thus, 48-6 signifies, that 48 is to be divided by 6. It is sometimes denoted by a horizontal line, (—) drawn under the 48 sig

dividend, and the divisor placed below the line; thus,

-

6

nifies, that 48 is to be divided by 6. It is likewise denoted by two vertical lines, or inverted parenthesis, ); thus, 6)48( signifies, that 48 is to be divided by 6.

NOTE.-Under this, as well as the other simple rules, the methods of proof introduced, are in themselves sufficient to illustrate the principles of the rules, and the relation which they bear to one another; and for the sake of becoming ac quainted with these principles, the student should as carefully examine every method of proof offered, as the rules them selves. Every difficulty will vanish before the student, every principle will appear easy and simple, and his progress will be rapid and interesting, if he will only attentively examine the illustrations and demonstrations given with the examples.

NOTE. The student will notice in the table, that the dividend is placed first, the divisor next, and the quotient or answer to each, at the right of the parallel lines. The student can make use of language similar to this, in repeating the table, 2 in 2 once, 2 in 4 twice, 2 in 6 three times; that is, 2 is contained, or can be subtracted from 2, once; and 2 is contained, or can be subtracted from 4, twice; &c.

Questions to be mentally answered by the Student. 1. If 10 oranges cost 20 cents; what will one cost?

2. If 3 apples cost 3 cents; what will I cost?

3. If 3 bushels of oats cost 6 shillings; what was that a bushel?