attention to these circumstances. At first, he should sum the Same column again and again, till he can add two or more figures at a glance. He should add without naming the figures, considering only the sums, as he runs up or down the columns. When the account is long, to prevent the trouble of going over the whole process again, in case of interruption or mistake, he should mark, on a separate piece of paper, or below the sum of each column, in a smaller character, the figure to be carried to the next column. EXERCISES.

1. Add together 2+4+3.

2. Add together 53+29.

3. Add together 34+23+87.

4. Add together 483+98 +956+85.

5. Add together 4753+6378+9257+2896.

6. Add together 729+853 +986+29+648+462. 7. Add together 974+65 +376 +487+598 +88. 8. Add together 4697 + 285 + 7496 + 3875 +914 +9283.

9. Add together 573+4136 +8767+846+9385 +1658.

10. Add together 295+89+376 +927+35+ 484 +973.

11. Add together 94+327+686 +475 +593 +887 +66.

12. Add together 9143 + 2596 +385 +49 + 4638 +9175+894.

13. Add together 856+9193+8765+ 4287 +6696 +9185+979.

14. Add together 263+97+835+ 469 + 538 +947TM +792+176+854.

15. Add together 7485+6394 +639+ 8476 + 3916 +2758+672 +875+4639.

16. Add together 5864+ 547 +9183749 +8857 +1798+7563+975+846.

17. Add together 546+679 +184+395+467 +938: +754+836+979+87.

18. Add together 173 +9469+ 837 +6528 +4275 +3678+9213+7534+5678+965.

19. Add together 967+245+96+873+7957+9547

+8698765+976+5389.

20. Add together 925 +678 +354+679 +438+567 +294+123+546+789+257 +985.

21. Add together 515+374

+797+835+569+273

+856+457+164+798+104+567.

22. How many days are there from 21st October to 30th April?

23. How many days are there from 15th May to 11th November?

24. There are in Scotland about 2,489,725 acres in grass; 140,095 acres in wheat; 280,193 in bar ley; 1,260,362 in oats; 500 in rye; 118,000 in pease and beans; 80,000 in potatoes; 407,125 in turnips; 16,500 in flax; 32,000 in gardens; and 218,950 acres in fallow: how many cultivated acres are there in Scotland?

SUBTRACTION.

SUBTRACTION is that process by which we find the difference between any two numbers. For distinction's sake, the greater of the two numbers is called the minuend, and the less the subtrahend.

RULE. Write the less number under the greater, placing units under units, tens under tens, and so on. Then, beginning at the right hand, subtract each figure in the lower line from the figure above it, and write the remainders in order, under the figures from which they proceed. If any figure in the lower line exceeds that above it, add 10 to the upper one, and subtract the lower from the sum, or rather subtract the lower figure from 10, and then add the upper figure to the remainder; in this case, you must add 1 to the next under figure, before you subtract it. The line of remainders thus obtained, exhibits the difference sought.

PROOF. Add the difference to the less number, and the sum, if the work is right, will be equal to the greater.

From 5385
Take 1794
Difference 3591

Proof

5385

EXAMPLE.

Illustration of the process. I take 4 from 5 and 1 remains; take 9 from 8 I cannot, I therefore borrow 10, and then I take 9 from 10 and 1 remains, which added to 8 are 9; I carry 1 to 7 are 8, take 8 from 3 I cannot, I therefore borrow 10, and take 8 from 10 and 2 re

main, which added to 3 make 5; I carry 1 to 1 are 2, take 2 from 5 and 3 remain.

REMARKS.

The Rule for Subtraction is founded on the three following axioms:

1. If every place in the minuend be greater than the corresponding place in the subtrahend, the difference of two numbers will be equal to the difference of all the units, plus the difference of all the tens, plus the difference of all the other places taken together: this is evident, because the whole difference must be equal to all the parts of that difference taken together.

2. Any place in the minuend may be made greater than the same place in the subtrahend, by adding 10 to it.

3. If equal additions be made both to minuend and subtrahend, the difference will be the same as before.

II. The process of subtraction may be carried on, like that of addition, either from right to left, or from left to right, and the result will be the same. The former method, however, is better adapted to practice.

III. When *,, or 2, is annexed to either or both of the given numbers, begin with subtracting the under fraction from that above it. When the upper fraction is the less of the two, borrow 4, from which subtract the under fraction, and to the remainder add the upper; write the sum, and carry 1 to the next under figure before you subtract it.

IV. Although it is usual to place the minuend above the subtrahend, yet it is sometimes more convenient to have the subtrahend uppermost; the learner should therefore acquire the habit of working either way with equal readiness.

EXERCISES.

1. From 4725367 take 2513143.
2. From 9214756 take 5192524.
3. From 543267819 take 51829247.
4. From 7285467 take 53985.

5. From 105274165 subtract 9623978.
6. From 12345678 subtract 8416397.
7. From 618537546 subtract 207812639.
8. From 92357944 subtract 23529183.
9. Subtract 39179123 from 56342047.
10. Subtract 669386 from 167534780.

*4 is one-fourth of 1, is one-half or two-fourths of 1, and is three-fourths

of 1.

11. Subtract 283519 from 1 million.
12. Subtract 3278463 from 100 millions.
13. Subtract 129173982 from 471087563.
14. Subtract 2708196 from 9104283.

15. Subtract 798003 from 2306051.

16. What is the difference between 61729286 and 84172865?

17. What is the difference between 468000000 and

9170865?

18. What is the difference between 28670157 and 46721086?

19. What is the difference between 782324089 and 29862179?

20. How much does 594105003 exceed 293300529 ? 21. How much does 99999 want of 1000000?

22. The first Logarithmical Tables were published in 1614, how long is it since?

23. Borrowed £514, of which I have paid £358, how much am I still due?

24. London is 393 miles from Edinburgh, and 272 miles from Newcastle, how far is Newcastle from Edinburgh?

25. From 450 26.6262 27.92 28.32

Take 2631 292

29.672 487

MULTIPLICATION is a short way of performing that particular case of Addition, in which all the numbers to be added are equal to one another; or, it is that process by which we find out the amount of one number called the multiplicand, when reckoned as often

as there are units in another number called the multiplier; the result of the operation is called the product; and the multiplicand and multiplier are called factors of the product.

1. When the multiplier does not exceed 12, write it under the unit's place of the multiplicand, and draw a line below it; then, beginning with units, multiply each figure of the multiplicand, one after another, by the multiplier, sctting down the right hand figure of each product, and carrying the rest to the following product. When you come to the last figure of the multiplicand, write down its complete product, with the carriage added to it.

2. When the multiplier exceeds 12, write it under the multiplieand, units under units, tens under tens, and so on; and draw a line below it; then multiply, as before, every figure of the multiplicand, by each of the figures of the multiplier, in any order; and place the first figure of each line of products directly under its respective multiplier. The sum of these several lines of products, is the product required.

Note 1. When there are ciphers intermixed with the figures of the multiplier, pass by them in the work altogether.

2. When the multiplier is 10, 100, 1000, &c. we have only to annex to the multiplicand one, two, three, &c. ciphers, respectively, for the product.

3. When there are ciphers annexed to either factor, or to both, place the factors so that the first digit of the multiplier, towards the right, may be under that of the multiplicand; then multiply the digits of the multiplicand by those of the multiplier, and annex the ciphers to the product.

PROOF. Add together the figures in each factor, casting out the 9's as they arise in summing, and multiply the remainders together'; then, if the excess of the 9's in this product be not equal to the excess of the 9's in the total product, the work is inaccurate.

Note. If an error of 9, or any of its multiples, has been committed in the work, or if a figure has been placed or added in a wrong column, the results will nevertheless agree in this method of proof But though it cannot be altogether depended upon, yet it is very useful, and more convenient than any other for practice. It should, however, be remembered, that as no method of proof can afford an absolute certainty of the accuracy of an arithmetical process, it is better to guard against errors, by using proper care in the computation, than to try to detect them after they have been committed.

*The learner should commit to memory the Multiplication Table before he proceeds farther.