Proof.

40. The proof of an operation, in Addition, consists in showing that the answer contains as many units as there are in all the numbers added. There are three methods of proof.

I. Begin with the units' column and add, in succession, all the columns in an opposite direction. If the work is right, the results will agree:

1

II. Divide the given numbers into parts, and add the parts separately: then add together the partial sums: if the work is right, the results will agree:

III. Find the excess of 9's in each number, and place it at the right (Art. 34). Add these numbers, and note the excess of 9's in their sum. This excess should be equal to the excess

of 9's in the sum of the numbers.

NOTE.-The third method of proof applies only to simple numbers.]

1. What is the sum of 182796, 143274, 32160, and 47047? and what the proof?

39. What is Addition? What is the sum of two or more numbers? On how many principles do the operations of Addition depend? What is the first principle? What the second? What the third? What the fourth? What is the Rule for Addition?

40. How many methods of Proof are there for Addition? What is the process in the first method? What in the second? What in the third?

41. What is the process of reading? How does it differ from spelling?

41. The pupil should be early taught to omit the intermediate words in the addition of columns of figures. Thus, in the above example, instead of saying, 7 and 0 are 7; 7 and 4 are eleven; 11 and 6 are seventeen; he should simply say, seven, eleven, seventeen. Then, in the column of tens, he should say, five, eleven, eighteen, twenty-seven; and similarly, for the other columns at the left. This is called reading the columns. Let the pupils be often practised in the readings, both separately and in concert in the class.

5. What is the sum of 1376, 38940, 8471, 23607, 891?

6. What is the sum of 3480902, 3271, 567321, 91243, 6001, 169?

7. What is the sum of 42300, 6000, 347001, 525, 47?

39. The population of the United States and Territories, in 1850, was as follows: White population, 19553068; Free Colored population, 434495; Slave population, 3204313; Indians, 400674: what was the entire population?

40. In the year 1850, the expenditures of the United States amounted to 43002168 dollars; in 1851, to 48905879 dollars; in 1852, to 46007893 dollars: what were the expenditures of the United States for these three years?

41. A man of fortune bequeathed to each of his three sons, 10492 dollars; to each of his two daughters, 5976 dollars; to his wife, the remainder of his property, which exceeded the amount bequeathed to his children by twelve hundred dollars : find the amount of his property.

42. A stage goes in one day 27 miles 3 furlongs 36 rods;

the next, 32 miles 10 rods; the next, 36 miles 2 furlongs; the next, 25 miles 6 furlongs 38 feet: how far did it go in 4 days?

43. Bought a barrel of flour for eight dollars and seventyfive cents; a ton of plaster for five dollars sixty-two and a half cents; a hat for three dollars twelve cents and five mills; fifty pounds of sugar for four dollars fifty cents and nine mills: what was the amount of my bill?

44. A lady bought a bonnet for $5.375; some silk for $12.03; some ribbon for $0.875; a shawl for $9.46: what did the whole amount to?

45. A wine-merchant taking an invoice of his liquors, finds that he has 5 hhd. 36 gal. 2 qt. of wine; 3 hhd. 15 gal. 1 qt. 1 pt. of rum; 1 hhd. 2 qt. of gin; 40 gal. 1 pt. of whiskey: how much liquor in all?

46. Tea was imported into the United States, in the year 1851, to the value of $4798005; in 1852, $7285817; in 1853, $8224853: what was the value of the tea imported during these three years?

47. The United States exported tobacco, in the year 1851, to the amount of $9219251; in 1852, $10031283; in 1853, $11319319: what was the entire value of tobacco exported in these three years?

48. A man sold his house and lot for $25840, which was $3186 less than he gave for them; how much did they cost him?

:

49. A speculator bought three city lots for the first he paid $2870.43; for the second, $2346.75; for the third, $1563.82. He sold the same at an average profit upon each of $476.25: what amount did he receive for the lots?

50. What is the fortune of a merchant who has $79650 in real estate, $25640 in merchandise, $9654 in furniture and library, $16835 in stocks, $12642 in debts due him, and $5685 in cash?

51. The churches of the United States and Territories, in 850, were: Baptists, 9375; Congregationalists, 1706; Presby