which is the same as dimes. When your subtrahend contains denominations which are not named in the minuend, what do you do? A. Supply the vacant denominations of the minuend with ciphers, then place the subtrahend under, so that cents may occupy the place of cents, and mills the place of mills. If you are required to subtract dimes from dollars, what do you first do? A. Join two ciphers to the right hand of the minuend, placing a separatrix between them and dollars; then place your dimes under the left hand cipher joined to the minuend, and to the dimes join a cipher, and they will stand as cents, and then subtract. How do you point off in subtraction of decimals? A. Place the separatrix in the result, directly below the separatrix in the given numbers. Why is federal money introduced under the head of decimals? A. Because dimes, cents and mills form a decimal expression of which a dollar is the integer.

SIMPLE MULTIPLICATION,

Is repeating one of two numbers as often as the other contains a unit; or, it is the shortest method of performing addition, where the same number is to be repeated a given number of times. The two given numbers are called multiplicand and multiplier. The multiplicand is the number to be repeated. The multiplier is the repeater, or number by which you multiply. The number produced by the operation of the work, is called the product. This is the most useful rule in practical arithmetick. When the price of one is given, by this rule, we obtain the price of any number, or quantity; when length and breadth are given, by it, we find the area or surface; in reduction, it affords the greatest facility in reducing higher denominations to lower, and its principles are advantageously applied in all practical business of buying and selling.

NOTE.-The multiplicand and multiplier taken together, are called factors or substitutes..

Multiplication is denoted by this character, X; thus, 6× 3-18, which signifies, that the product of 6 multiplied by 3, is 18.

NOTE.-No pains should be spared by the student in making himself master of the following table. The task is easy if persevered in ; but when the student suffers himself to pass over it superficially, no time afterwards spent in work is scarcely sufficient to make it familiar to his mind. A good knowledge of it will greatly facilitate his progress, and save him the trouble of repeatedly reviewing the same work to detect mistakes.

The student should be required mentally to answer the fol

lowing questions.

What will 3 apples come to, at 3 cents each?

What will 8 bushels of wheat cost, at 8 shillings a bushel? 8 times 8 are how many?

What will 10 bushels of oats come to, at 2 shillings a bushel?

What are 9 yards of calico worth, at 3 shillings a yard? 3 times 9 are how many?

What must you pay for 12 cows, at 12 dollars each?

What will 4 cows come to, at 10 dollars each? What will 5 cows? what will 6 cows? what will 7 cows? what will 8 cows? what will 10 cows?

What will 11 ploughs cost, át 11 dollars each?
What will 9 hogs cost, at 9 dollars each?

A pound contains 20 shillings; how many shillings in 2 pounds? in 3 pounds? in 4 pounds? in 5 pounds? How many are twice 20? how many are 3 times 20? 4 times 20? 5 times 20?

Twelve pence make 1 shilling; how many pence in 2 shillings? in 3 shillings? in 4 shillings? in 5 shillings? in 6 shillings? in 7 shillings? in 8 shillings? in 9 shillings? in 10 shillings? in 11 shillings? in 12 shillings? How many are twice 12? 3 times 12? 4 times 12? 5 times 12? 6 times 12? 7 times 12? 8 times 12? 9 times 12? 10 times 12? 11 times 12? 12 times 12?

CASE I-When the multiplier is a single, significant figure. RULE.-Place the multiplier under the right hand figure of the multiplicand, and draw a line underneath. First, multiply the right hand figure of the multiplicand by the multiplier; and when the product does not exceed 9, place it directly under; but if the product exceed 9, place down the right hand figure of the product; and add the left to the product of the next figure of the multiplicand, and so proceed, till you have multiplied all the figures of the multiplicand by the multiplier; remembering to set down the whole product of the left hand figure.

METHODS OF PROOF.-1st. Make the multiplicand a multiplier, and if it produce the same result, the work is right.

2nd. Multiplication may be proved by addition.

Write the multiplicand down as many times as the multiplier expresses a unit; then add, and if the same result be produced, the work is right.

3d. Multiplication may be proved by subtraction.

From the product, subtract the multiplicand as many times, as the multiplier expresses a unit; and if it diminish it to nothing, the work is right.

4th. Lastly, multiplication may be proved by division.

Divide the product by either of the factors, and if the operation produce the other, the work is right. Although this is the best method

of proof, yet it should be omitted, till the student has become acquainted with division.

Other methods of proof might be given; but these are introduced on account of their being the most simple, and best suited to our purpose in illustrating the principles of this rule; and showing the relation which it bears to the other simple rules.

1. Multiply 8 by 4. Multiplicand, 8 Multiplier,

4

Product, 32

EXAMPLES.

We place the multiplier under the multiplicand, as our rule directs. Then we say 4 times 8 are 32, placing the product under; we then have 32 for a product, which is 8, 4 times repeated.

DEMONSTRATION.-That this is a short way of performing addition, Is very evident: for we arrive at the result at once, which, in addition, requires the multiplicand 8, to be set down four times and added; thus, 8+8+8+8=32.

8

1st. EXAMPLE.-Proved according to the first method. Multiplicand, 4 What was before the multiplicand, Multiplier, now becomes the multiplier. We now say 8 times 4 are 32, the same product Product, 32 as was produced by the first operation. DEMONSTRATION.-It is plain, that it can make no difference, whether 8 be repeated 4 times, as, 4 times 8 are 32; or 4 repeated 8 times, as, 8 times 4 are 32, the same result is produced.

1st. EXAMPLE.-Proved according to the 2nd method. DEMONSTRATION. Here, we set down the multiplicand 4 times because in the example, our multiplier expresses

8

8

units; we then add, and the result clearly demonstrates, that when the same number is repeated by multiplying, it may be repeated by adding; and consequently proves the work. But proof by addition is a more tedious method than proof by multiplica32 Amount. tion, and is therefore only introduced to show the relation between the two rules. . It is also plain, that it can make no difference, whether we add the multiplicand as many times as our multiplier expresses a unit; thus, 8+8+8+8=32; or add the multiplier as many times as our multiplicand expresses a unit, thus, 4+4+4+4+474+4+4=32; the same result, you perceive, must be produced.

1st. EXAMPLE.-Proved according to the 3d method. Multiplicand, 8

DEM. You have just learned, that multiplicaMultiplier, 4 tion is a short way of performing addition.. You Carried up.

Product,

- Brought up.

24

32 also understand, that subtraction is made to prove
8 addition; consequently it may be made to prove
for subtraction is the reverse of ad-
multiplication;
8 dition, and 32, (the product in our example,) is 8,
-four times expressed; then if 8 be taken away from
32, (the product,) 4 times, it must evidently dimin-
ish it to nothing; because it is taking away 8, the
8 multiplicand, as many times as it has been repeated
8 by 4, the multiplier.

16

8

1st EXAMPLE.-Proved according to our 4th method. DEMONSTRATION.-Division being exactly the reverse of multiplica

4

8324
>32

tion, is consequently made to prove it; because, when we multiply 8 by 4, the 8 is 4 times repeated; and when this 8 is made a divisor, we find that 32 contains it 4 times, which gives us our other factor; and repeating the 8 by this factor, it again produces 32, the same as our first product; which must always be the case; because, it is only repeating the same number a second time, and must produce the same product; and when we come to subtract this product, we can have no remainder; because, taking the same number from itself can leave no remainder.

2. What will five yards of broadcloth cost, at three dollars a yard?

Proof by Addition.

3 Multiplicand.
5 Multiplier.

$15 Ans.

Proof by
Multiplication.

10.00

5

3

15

44

Here, by multiplying the 3 by 5, we repeat the 3 five times; and it is evi dent, that five yards are worth five times. as much as one. If we were not acquainted with multiplication, we would be under the necessity of setting 3 down five times, and adding as our proof shows.. From this sum, we learn, that when the price of one is given; as 1 yard, 1 pound, 1 ounce, &c. we may obtain the i price of the quantity, by multiplying the price of a unit, by the quantity; for the quantity when made a multiplier is considered to be a number containing as many units as the quantity contains yards, pounds, ounces, &c.

3. What will 9 calves come to, at 3 dollars each?

Ans. $27. 4. What is the worth of 9 bushels of clover seed, at 9 dollars a bushel?

Ans. $81