to add? Why? Why are the tens of one column carried as units to the next column to the left? What is an integer? See p. 15. What is a fraction? What is a decimal fraction? Are decimal fractions added differently from integers? Are the tens carried from the column of tenths to that of units in the same manner as from one rank of integers to the next? Why? To what rank does the right hand column belong, in a number consisting wholly of integers? Is this the case when a number consists partly or wholly of decimal fractions? How, then, is the local value of the figures ascertained? Explain the two methods of proving addition.

Practical Exercises.

1. By a census taken in the year 1841, the population of the British Islands was as follows: England, 14,995,138; Wales, 911,603; Scotland, 2,620,184; Ireland, 8,175,124; army, navy, &c., 193,469; islands in the British seas, 124,040. What is the total population of these islands?

Ans. 27,019,558.

2. In the same year, the number of emigrants that left the United Kingdom of Great Britain was as follows: to their North American colonies, 38,164; to the United States, 45,017; to the Australian colonies and New Zealand, 32,625; to all other places, 2786. What was the whole number of emigrants from the kingdom that year? Ans. 118,592.

3. The value of the principal articles of manufacture in Great Britain for that year is estimated as follows: cotton, £35,000,000 ;* woollen, £24,000,000; iron and hardware, £20,000,000; watches, jewelry, &c., £3,000,000; leather, £13,500,000; linen, £8,000,000; silk, £10,000,000; glass and earthenware, £4,250,000; paper, £2,000,000; hats, £2,000,000. What was the gross amount in pounds sterling? Ans. £121,750,000.

4. The white population of the United States, according to the census of 1840, classed according to ages, was as follows: Males, under five years of age, 1,270,790; of five and under ten, 1,024,072; of ten and under fifteen, 879,499; of fifteen and under twenty, 756,022; of twenty and under thirty, 1,322,440; of thirty and under forty, 866,431; of forty and

*This character, £, stands for pounds in money.

under fifty, 536,568; of fifty and under sixty, 314,505; of sixty and under seventy, 174,226; of seventy and under eighty, 80,051; of eighty and under ninety, 21,679; of ninety and under a hundred, 2,507; of a hundred and upwards, 476. Females, under five, 1,203,349; of five and under ten, 986,921; of ten and under fifteen, 836,588; of fifteen and under twenty, 792,168; of twenty and under thirty, 1,253,395; of thirty and under forty, 779,097; of forty and under fifty, 502,143; of fifty and under sixty, 304,810; of sixty and under seventy, 173,299; of seventy and under eighty, 80,562; of eighty and under ninety, 23,964; of ninety and under a hundred, 3,231; of a hundred and upwards, 315. What was the number of free white males? Of free white females? Total of free whites? Ans. to the last question, 14,189,108.

5. The number of free people of color in the United States, according to the census of 1840, was as follows: Males, under ten years of age, 56,323; of ten and under twenty-four, 52,799; of twenty-four and under thirty-six, 35,308; of thirtysix and under fifty-five, 28,258; of fifty-five and under a hundred,13,493; of a hundred and upwards, 286. Females, under ten, 55,069; of ten and under twenty-four, 56,562; of twenty-four and under thirty-six, 41,673; of thirty-six and under fifty-five, 30,385; of fifty-five and under a hundred, 15,728; of a hundred and upwards, 361. What was the number of free males of color? Of free females of color? Of free persons of color?

Ans. to the last question, 386,245.

6. The number of slaves in the United States, according to the census of 1840, was as follows: Males, under ten years of age, 422,599; of ten and under twenty-four, 391,131; of twenty-four and under thirty-six, 235,373; of thirty-six and under fifty-five, 145,264; of fifty-five and under a hundred, 51,288; of a hundred and upwards, 753. Females, under ten, 421,470; of ten and under twenty-four, 390,075; of twentyfour and under thirty-six, 239,787; of thirty-six and under fifty-five, 139,201; of fifty-five and under a hundred, 49,692; of a hundred and upwards, 580. What was the number of male slaves? Of female slaves? Of slaves of both sexes? Ans. to the last question, 2,487,213.

7. What was the whole population of the United States by the census of 1840, including 6100 persons of the navy of the United States, who were not reckoned in either of the last three enumerations? Ans. 17,068,666.

8. The following table exhibits the territorial area of the states of Europe and of North America. Find the aggregate, which will be correct if it amounts to 11,658,480.

AREA OF EUROPE.

Russian Empire, including Poland

Austrian Empire, including Lombardy, &c.

Square miles.

2,000,000

257,368

France, including Corsica

203,736

9. The following is a statement of the white population of each of the states and territories of the United States and of the District of Columbia, according to the census of 1850, followed by tables of the free colored and slave population for the same period. Find the sum of each of the columns of males and females, which will be correct if they agree with the several given aggregates. Then take the sum of the aggregates, which will be correct if it amounts to 23,190,998.