ADDITION.

25. Simple Addition⚫ teaches to collect two or more whole numbers into one, which is called their sum.

The mark for Addition is + plus, (more,) and shews that the number which follows the sign, is to be added to the number standing before it.

26. To add single figures together.

RULE I. Begin at the bottom, and find what number will

■ The word Addition is derived from the Latin addo, to put to; sum from summa; and proof from probo, to prove, or make out.

b This rule depends on the method of notation; (Art. 12.) thus, in the first example, if I want to know the sum of 7 and 4, I must evidently resolve each into the units of which it is composed, and then count all the units in both, one by one, to find the amount; and this practice I must follow until my mind acquires from habit, a sufficient dexterity in numbering to do without it. The only method that a person totally ignorant of Addition could employ, would be this; he would write down all the ones in each of the numbers to be added, and then count the whole. Thus ex. 1. would stand according to such a method,

These ones being counted, are found to amount to 21, which is the sum of the given numbers. Simple as this explanation may appear, it is plain that the reason of the rule can be shewn on no other principle. By this method this following Table was first calculated.

and 5 in the top line, and at the point where the two rows of figures (one vertical, the other horizontal) meet stands 9, which is the sum of 4 and 5. To find the sum of 7 and 6, look for 7 on the left, and 6 at top, and at the point where the lines containing 7 and 6 meet stands 13, their sum. In like manner 5 and 9 are found to be 14; 8 and 3 are 11; 9 and 9 are 18, &c. &c. Those who

arise by taking the units in the lower figure, and the units in the next figure above it, into one sum.

II. Do the same with this sum and the third figure-with this last sum and the fourth figure, and so on until all the figures have been taken; set down the last sum between two lines below, and it will be the sum required.

Method of proof. Draw a line under the top figure; then add up all the rest of the figures as before, and place the sum under the former sum; add this last sum and the top figure together, and if the sum is the same as the sum first found, the work is right. EXAMPLES.

1. Add the figures 3, 2, 5, 4, and 7 together.

OPERATION.

3

5

4

7

Sum 21 18

Proof 21

Explanation.

I first place the given figures in a column under one another; then, beginning at the bottom, I say, 7 and 4 are 11, then 11 and 5 are 16, then 16 and 2 are 18, then 18 and 3 are 21, which (because all the figures have been used) is the sum; I therefore place it at the bottom. Next I cut off the upper figure 3, and, beginning at the 7, I add all up as before, except the 3 cut off, and place the sum 18 below the former. Then I add the last sum 18 to the 3 cut off, and the sum is 21, which being the same as the sum first found, shews that the work is right.

26 B. To add any whole numbers together.

RULE I. Place the numbers under one another, so that units may stand under units, tens under tens, hundreds under hundreds, &c.

II. Add up the figures in the units (or right hand) column by the former rule; take out all the tens from the sum, and set

cannot readily add small numbers, ought to learn this Table by heart; thus the method of adding small numbers being once familiar, that of adding larger numbers will be gradually acquired by practice.

down (below the figures added) what is over, or, if nothing be over, set down a cipher.

III. Carry as many units (or ones) to the second column as there were tens in the first; add these up with the second column as before; take out the tens from the sum, set down the remainder, carry 1 for every ten to the third column, and proceed in this manner till the left hand column is added, under which its whole sum must be put down *.

3. Add 312, 498, 387, 968, and 527 together.

OPERATION.

312

498

387

968

577 Sum 2742

2433

Proof 2742

Explanation.

Having placed the numbers, I find that the sum of the units column is 32, or 3 tens and 2 over; I put down 2, and carry 3 to the second colunin, the sum of which is 34; I therefore put 4 down, and carry 3 to the last column, the sum of which is 27, which I put down. The second line and proof are done as directed in the last rule.

Having explained the method of adding, it remains to account for the method of carrying prescribed in the rule. Thus, in example 3, the sum of the units is 32, or 3 tens and 2 units; I must evidently put down the 2 units; but it is plain that the 3 tens must be added with the tens, namely, with the second column, which consists of tens: again, the sum of the second column (with the 3 carried) is 34, that is 34 tens, or 340; the 4 tens then must evidently be put down under this second column, (which is tens,) and the 3, which are hundreds, must be collected with (or carried to) the hundreds, the sum of which is 27 hundreds, or (which is the same) 2 thousand 7 hundred; this sum, it is plain, must be put down, as there can be no further carrying. Thus the rule teaches not only to collect several numbers into one, but likewise to class and arrange the different denominations in the sum, by continually reducing lower to higher denominations, as often as a sufficient number of the former arises.

The method of proof in this and the foregoing rule will be easily understood; for having cut off the top line, and added up all the rest of the figures, the result will be the sum, exclusive of the top line; wherefore if the said result and top line be added together, the number thence arising will evidently be the same as the sum, or upper line of the work.

9. Add the numbers 4321, 8037, 2345, 6728, and 1091 together. Sum 22522.

10. Add 109, 1237, 34, 987, and 12 together. Sum 2379. 11. Add nine thousand eight hundred and sixty-seven to the sum of the following numbers, 98, 876, 129, 9086, and 12345. Sum 32401.

12. A has 39 marbles, B has 68, C 24, D 190, E 59, and

F 95; how many have they among them? Ans. 475 marbles.

13. Received of G 12 shillings, of H 45, of K 130, of L 679, of M 99, and of N nine hundred and ninety-nine; how many did I receive in all? Ans. 1964 shillings.

14. A person to maintain himself and family 5 years spent as follows; viz. the first year 6871. the second, 9891. the third, 8361. the fourth, 10941. and the fifth, 12091. what did he spend in all ? Ans. 48157.

SUBTRACTION.

27. Simple Subtraction teacheth to take a less whole number from a greater, whereby the remainder or difference is known. The mark for Subtraction is -; it is named minus, (or less,) and shews that the number following the sign is to be taken from the number which stands before it.

RULE I. Place the less number below the greater, and let units stand under units, tens under tens, &c. as in Addition, and draw three lines at proper intervals below.

II. Begin at the right hand figure in the less number, and take

d. The name Subtraction comes from the Latin sub under, and traho to draw.

it out of the figure above, and set what remains under it: do the same with all the figures in the less number, setting each remainder under the figure from whence it arises.

III. But if it happens that a figure in the lower line be greater than that above it, add ten to the upper one, after which take the lower figure from the sum; set down the remainder, and carry one to the next lower figure before you subtract.

IV. Proceed in this manner until all the lower figures are subtracted, and the result will be the remainder, or difference required.

Method of Proof. Add the difference found and the less number together, then if the sum be equal to the greater number, the work is right".

• To shew that this rule has its foundation in Notation, let it be required to take 3 from 7; now if we represent the 7 by its units, agreeable to art. 12, and cut off from these the number of units in 3, the rest of the units being counted will shew what remains, thus, 1,1,1|1, 1, 1, 1, where having cut off 3, the rest of the units being counted I find amount to 4; therefore 3 taken from 7, 4 remains; and the same may be shewn of other numbers.

When each figure in the lower line is less than its correspondent figure in the upper, it is plain, that by taking each lower figure from that above it, we obtain the several differences of all the parts, which, taken together in order, will evidently constitute the difference of the whole.

But when any figure in the upper line is less than its correspondent one in the lower, we borrow 10, which is evidently 1 of the next higher denomination, after which we carry 1 (to make up for the borrowing) to the lower figure of the next higher denomination; by thus increasing the number to be subtracted, we eventually take the 1 away from that denomination from whence it had before been nominally borrowed.

For the method of proof. If a less number be taken from a greater, what remains will be the difference; and if the difference of these two numbers be added to the less, the sum will be the greater: this is too plain to require illustration; we will barely apply it to the first example, where the upper line is the greater number, the second line the lesser, the third line the difference, and the fourth line (which is equal to the greater) is the sum of the difference and less number.

The Table in the note on art. 26. may be made a Subtraction Table: thus to find the difference of two numbers, look for the least at top, and in the same column find the greatest; then the number in the left hand column, which stands in the same line with the greatest, is the difference required: thus to take 7 from 12, look for 7 at top, and 12 in the same column, then opposite 1-2 in the left hand column stands 5, the difference.