1 and 7 are 8, and 4 are 12. I put down the whole sum, 12 thousands, because there are no more columns to be added. Teacher. What is the sum of all the columns? Pupil. Twelve thousand seven hundred and twenty-three.

After a short time, the pupil should learn to sum the several columns without naming the figures. For instance, in the above example, instead of saying 5 and 6 are 11, and 1 are 12, and 7 are 19, and 4 are 23, he should only be required to name the result at each step; thus-5, 11, 12, 19, 23, &c.

REASON OF THE RULE.-The rule for adding numbers depends on the following obvious principles:

1. That the whole sum is equal to all its parts taken together.

2. That it is only numbers of the same local value that can be added together; hence, units are added to units, tens to tens, &c.

3. One is carried for every ten, because, from the nature of notation, ten units in any one column is equal to one only in the column immediately to the left of it.

PROOF.-To check the correctness of the addition, count the columns downward, and if the sum is the same as before, the work is probably right.

Or, Divide the account into two or more parts, and add the parts separately; then find the amount of these partial sums. If this amount is equal to the sum obtained in the usual way, the work may be presumed to be right.

51. 3917 +46 + 31254 + 293 +8162 + 753 +82 + 7854. 52. 17542 + 39126 +3175 +93 +814+79268 + 123 + 17. 53. 891 +216 + 7139 + 25 + 318 + 9832 + 72956 + 18 +9.

Answers.

31. 46672 36. 44810 41. 31631
32. 40948 37. 41058 42. 38630
33. 32239 38. 45285 43. 38499
34. 28426 39.30969
35. 39118 40. 43263

44. 34643

46. 452745 51. 52361 47. 5379631 52. 140158 48. 5030204 53. 91404 49. 4157698

45. 383034 50. 5301997

54. Add together fifty-four thousand seven hundred and sixteen, four thousand and eighty-nine, three hundred and twenty thousand six hundred and four, one hundred and sixty-five, thirty-nine, four millions three thousand two hundred and thirtyseven, nine hundred and sixty-one thousand five hundred and twenty-nine.

55. In 1851, the population of London was 2362236; that of Liverpool, 375955; of Manchester, 303385; of Birmingham, 232841; of Leeds, 172270; of Bristol, 137328; of Sheffield, 135310; of Bradford, 103778; of Glasgow, 329097; of Edinburgh and Leith, 191221. What was the total population of all these places in that year?

56. In 1801, the population of London was 958863; that of Liverpool, 82295; of Manchester, 94876; of Birmingham, 70670; of Leeds, 53162; of Bristol, 61153; of Sheffield, 45755; of Bradford, 13264; of Glasgow, 77058; of Edinburgh and Leith, 81404. What was then the total population of all these places?

57. The Great Exhibition of 1851 was open in May 27 days, and the number of visitors during that time was 734814; in June, 25 days-visitors, 1133114; in July, 27 days-visitors, 1314176; in August, 26 days-visitors, 1023435; in September, 26 days-visitors, 1155240; and in October, 13 days-visitors, 808237. How many days was the Great Exhibition open, and how many visitors were in it?

58. At the end of 1843 there were open 2036 miles of railway in Great Britain; in 1844 there were opened 204 miles; in 1845, 296 miles; in 1846, 606 miles; in 1847, 803 miles; in 1848, 1182 'miles; in 1849, 869 miles; and in 1850, 625 miles. How many miles of railway communication were open at the end of 1850 ?

59. The total receipts for all the railways in Great Britain were as follows: in 1842, £4341781; in 1843, £4842625; in 1844, £5610982; in 1845, £6669224; in 1846, £7689874; in 1847; £8975671; in 1848, £10059006; in 1849, £11013817; and in 1850, £12755235. What were the total receipts of all the railways during these nine years?

60. In the year 1857 the quantity of mineral dug in the United Kingdom was as follows: tin ore, 9783 tons; copper ore, 229455 tons; lead ore, 96820 tons; silver, 15 tons; zinc ore, 9289 tons; sulphur ore, 74679 tons; arsenic, 476 tons; cobalt, 4 tons; nickel, 1 ton; iron ore, 9573281 tons; coals, 65394707 tons; salt, 1462045 tons. How many tons were dug in all?

SIMPLE SUBTRACTION.

SUBTRACTION is the method of taking a less number from a greater, to find what remains, or what is the difference between the two numbers.

The number left after subtracting the one from the other, is called the Difference, or Remainder.

Subtraction is denoted by the sign called minus; thus, 7-52, denotes that 5 is to be subtracted from 7, and the remainder is equal to 2.

RULE FOR SUBTRACTING.

1. Write the less number under the greater, so that units may stand under units, tens under tens, &c., and draw a line under them.

2. Begin at the right hand, and take in succession each figure in the lower line, if possible, from that which stands above it in the upper line, and set down the remainder.

3. But if any figure in the lower line be greater than the figure above it, 10 is to be added to the upper figure before subtracting.* As an equivalent for adding the 10, the next under figure requires to be considered as 1 more. This is called carrying I to the under figure.

The several remainders that have been written down form the answer, or difference between the two lines.

The 10 that is here added is got by taking, or borrowing, as it is called, 1 from the next upper figure. That figure should, therefore, be counted as one less; but it is more convenient in practice, and produces the same result, to make the next under figure 1 more. The reason that 1 taken from the one figure is counted as 10, in adding it to the other, is, that the figure from which the 1 is taken is of a higher rank than the figure to which it is added; and, consequently, 1 of the former is equal to 10 of the latter.