But the usual method of proving Addition is to begin at the upper line and add downwards, in the same manner as it was added upwards, then if the sums agree, we may conclude the work is right.

SIMPLE SUBTRACTION.

8. SIMPLE SUBTRACTION is the operation of taking a less number from a greater, or finding the difference of two proposed numbers: thus, 1 btracted from 7 leaves 6, which is the difference of 1 and 7; 8 subtracted from 10 leaves 2, the difference of 8 and 10; 22 subtracted from 33 leaves 11 the difference; for 2 units taken from 3 units leaves 1 unit; and 2 tens taken from 3 tens leaves 1 ten; therefore 1 ten and 1 unit, or 11 is the difference. And hence it is evident that in placing numbers for subtraction, units must stand under units, tens under tens, hundreds under hundreds, &c. as in addition.

Ex. 1. From 33

Take 22

Difference or remainder 11

9. The method of proving subtraction is to add the less number and the difference or remainder together, for their sum must evidently be equal to the greater number if the work is right: thus, let the difference of 4356 and 3213 be required.

10. When the figure to be subtracted is greater than that directly above it, the method of operating is easily derived

thus:

Let the difference of 41 and 18 be required:

41

18

differ. 23; here 8 cannot be subtracted from 1, but if 10 is taken from the 40 and added to the 1 the sum is 11, then 8 from 11 and 3 remains; consequently the 1 which stands under the 4 must be subtracted from 3 (or 4 lessened by 1), and the remainder is 2. In like manner proceed with any other number of figures.

Ex. 4. From 823
Take 636

Rem. 187: here 6 from 13 (10 added to 3) and 7 remains; 1 from 11 (10 added to 2 lessened by 1) and 8 remains; 6 from 7 (3 lessened by 1) and 1 remains. But it evidently comes to the same thing if we augment the lower figures by 1 instead of lessening the upper figures; thus 6 from 13 and 7 remains; 4 from 12 and 8 remains; 7 from 8 and 1 remains.

Ex. 5. From 14040
Take 3051

Rem. 10989; here 1 from 10 and 9 remains; 5 from 13 (10 added to 4 lessened by 1) and remains; 0 from 10 lessened by 1 and 9 remains; 3 from 4 lessened by 1 and 0 remains: lastly as there is nothing to subtract from the 1, it becomes the left-hand figure of the remainder.

If we augment the lower figures by 1 instead of diminishing the upper ones, the process will be thus: 1 from 10 and 9 remains; 6 from 14 and 8 remains; 1 from 10 and 9 remains; 4 from 4 and 0 remains.

11. Or subtraction may be performed by setting down such figures for the remainder that when added to the less number shall give the greater.

Thus, from 9875

Take 2301

Rem. 7574; here 1 and 4 make 5, therefore 4 is the remainder; 0 and 7 make 7 for the remainder; 3 and 5 make 8, therefore 5 is the remainder; 2 and 7 make 9, therefore 7 is the remainder.

When the lower figure is greater than that directly above, it is evident that the next lower figure must be augmented by 1.

Thus, from 10126
Take 1357

Rem. $769; here 7 and 9 make 16, therefore 9 remains; 6 (or 5 augmented by i) and 6 make 12, therefore 6 remains; 4 (or 3 augmented by 1) and 7 make 11, therefore 7 remains; 2 (or 1 augmented by 2) and 8 make 10, therefore 8 remains.

SIMPLE MULTIPLICATION.

12. SIMPLE MULTIPLICATION consists in finding the sum or amount of a proposed number taken or repeated a given number of times, and may be denominated a compendious method of Addition: for example, suppose 6 is to be taken 3 times :

6

6 6

then the addition gives 18, but by multiplication we say 3 times 6 make 18.

The number to be multiplied is called the multiplicand; that by which you multiply, the multiplier; and the result is

called the product. The multiplicand and multiplier are without distinction called the terms or factors of the multiplication, because they make the product or number sought: thus 3 times. 5 make 15.

13. But in the first place it will be necessary to learn perfectly the following Table, which contains the products of every two of the 9 digits.

MULTIPLICATION TABLE.

To find the product of two figures in this table, look for one of them in the left-hand column, and for the other at top, then their product will be found where the vertical column from the top intersects the horizontal one from the left. Let 6 and 7 be proposed, then the columns meet at 42; for 6 times 7, or 7 times 6 make 42.

14. The rule for multiplying by a single figure is derived from addition; thus: Let the sum of 3 times 875, or, which is the same thing, the product of 875 by 3, be required?

To perform the addition; 5, 5, and 5 make 15, or 5 more than 1 ten; 7, 7, and 7 make 21, and 1 make 22, or 2 more than 2 tens; next 8, 8, and 8 make 24, and 2 make 26. But in the multiplication we say 3 times 5 make 15, or 5 more than 1 ten; 3 times 7 make 21, and 1 make 22, or 2 over 2 tens; lastly, 3 times 8 make 24, and 2 (for the 2 tens) inake 26. Therefore in multiplication, 1 must be carried to the left for every 10 in the products, and the overplus set down as in addition.

15. When the multiplier consists of one figure with ciphers. on the right, multiply by that figure, and annex the ciphers to the right of the product.

Ex. 4. Multiply 11

By 300

Product 3300 this is evident from Notation.

When the multiplier consists of several figures.

16. Begin at the right, and multiply by each figure separately, and set down the products so that the units of the second line may stand under the tens of the first, the units of the third line under the tens of the second, and so on: then add all the products together for the amount.

The reason for setting down the products by the single figures in this manner will be manifest, if we consider that the whole amount must (in the present example) consist of 3 times 231, 20 times 231, and 300 times 231, when added together:

Sum 74613. Here if the ciphers are cancelled (as having no value in the addition) the first figure of any line, or product by a single figure, must necessarily fall one place to the left of that above it. And hence the rule for multiplying by several figures is deduced.

17. When ciphers are between other figures in the multiplier, neglect them, remembering to set down the lines of products as far to the left as they would be if the ciphers were others figures.