eighty-eight, equal to four pounds, eight shillings. The eight is set down under the column of shillings, and the four is carried on to the column of pounds, the figures of which are added and placed as in simple addition.

The surest proof of the justness of any operation of addition, is to begin at the top of each column, and add downwards. If the amount be the same, it may be concluded that the operation is just.

Subtraction teaches how to find the difference between any two numbers by taking the less from the greater.

Rule. Place the less number under the greater, in the same manner as addition; begin at the unit column, and continuing from the right to the left, subtract each figure from that which stands over it, setting down the remainder under, and a cypher when nothing remains. Thus,

From 9876543
Take 7765432

Remainder 2111111

But if any figure in the lower line be greater than the figure above it, add ten to the upper figure; from their amount, subtract the lower figure, carry one to the next under figure, and continue as before; thus,

From 87123172

Take 18234433

Remainder 68888739

For a proof, add the remainder to the lower line of figures, and if the amount be the same as the upper line, the operation is right; as,

18234433

68888739

87123172

In the former of the preceding examples, begin with the unit figure, three; and as that cannot be subtracted from two, the figure above it, borrow one ten from the seven tens which stand next the two, and add that one ten to the two, which makes it twelve; then subtract the three from that compounded number twelve, and there remains nine. Set down the nine under the three, and proceed. But as one ten has been borrowed from the tens, there will remain but six tens instead of seven, from which six, when three, the under figure is subtracted, there will remain three, which set down below; and go on as before. Or, when one ten is thus borrowed, it may be added to the lower figure, instead of being subtracted from the upper.

RULE FOR COMPOUND SUBTRACTION. As in simple subtraction, place the smaller under the greater numbers, and subtract each number in the lower line from the corresponding figure in the upper, and set down the remainders. When any one of the lower numbers is greater than the number above it, add to the upper number, as many as make one of the next superior denomination; from which com

bined numbers subtract the figure in the lower line, set down the difference, and carry one to the next number in the lower line; which subtract as before. Example.

MULTIPLICATION is the method of finding the amount of any given number, repeated a given number of times. It is a kind of addition, but much more expeditious than the common mode of addition. The number to be multiplied or repeated, is called the multiplicand, the multiplying number is called the multiplier, and the result of the operation is called the product. Both the multiplier and multiplicand are termed factors.

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Rule. When the multiplier does not exceed twelve, begin at the units' place of the multiplicand; and if the product do not exceed nine, set it down under the multiplier; but if it be ten, or more, set down only the units, and carry the tens to be added to the next product, and let the last product be set down entire.

Proof.

Write down the figures of the multiplicand as many times as they are contained in the multiplier, and add them together.

When the multiplier consists of several figures, multiply the multiplicand successively, by each figure in the multiplier, placing the first figure of each product exactly under that figure of the multiplier, by which you are multiplying. Then add the columns of products.

Example -
Multiplicand
Multiplier

27831

243

83493

111324

55662

6762933

Product

Proof. Multiply the multiplier by the multiplicand.

Rule for multiplying given numbers of different denominations; or compound multiplication. Multiply the number of the lowest denomination by the multiplier. If the product be less.

than will make one of the next higher denomination, set it down; but if greater, find how many of the next higher denomination it contains; carry them as in addition, and write down only the remainder.

If the multiplier exceed twelve, multiply by its component parts successively; but if the multiplier cannot be exactly produced by the multiplication of simple numbers, take the nearest number to it, either greater or less, and multiply by its parts as before. Then multiply the given multiplicand by the difference between the assumed number and the multiplier, and add the product to that before found, when the assumed number is less than the multiplier; but subtract the same when it is greater.

Examples.

DIVISION teaches in what manner to find how often one number is contained in another number,.and is a short method of performing sub