hundreds. Add the column of hundreds and write down the sum, and the entire sum is 1535.

2. When the sum, in any column, exceeds 9, it produces one or more units of a higher order, which belongs to the next column at the left. In that case, write down the excess over exact tens, and add to the next column as many units of its own order, as there were tens in the sum.

This is called carrying to the next column. The number to be carried, should not, in practice, be written under the column at the left, but added mentally.

Hence, to find the sum of two or more numbers, we have the following

RULE.

I. Write the numbers to be added, so that units of the same order shall stand in the same column. 2

II. Add the column of units. Set down the units of the sum and carry the tens to the next column.

III. Add the column of tens. Set down the tens of the sum and carry the hundreds to the next column; and so on, till all the columns are added, and set down the entire sum of the last column.

PROOF.

34. The proof of any operation, in Addition, consists in showing that the result or answer contains as many units as there are in all the numbers added, and no more. There are two methods of proof, for beginners.*

I. Begin at the top of the units column and add all the columns downwards, carrying from one column to the other, as when the columns were added upwards. If the two results agree the work is supposed to be right. For, it is not likely that the same mistake will have been made in both additions.

II. Draw a line under the upper number. Add the lower numbers together, and then add their sum to the upper numbers/

*NOTE.-If the teacher prefers the method of proof by casting out the 9's, that method, for the four ground rules, will be found in the University Arithmetic.

34. What does the proof consist of in addition? How many methods of proof are there? Give the two methods.

NOTE.—Explain the process of addition by reading the figures.

If the last sum is the same as the sum total, first found, the work may be regarded as right.

EXAMPLES.

1. What is the sum of the numbers 375, 6321, and 598?

The small figure placed under the 4, shows how many are to be carried from the units' column, and the small figure under the 9, how many are to be carried from the tens' column.

OPERATION.

375

6321

598

7294

11

Also, in the examples below, the small figure under each column shows how many are to be carried to the next column at the left. Beginners should set down the numbers to be carried, as in the examples.

The pupil should be early taught to omit the intermediate words in the addition of columns of figures. Thus, in example 22, instead of saying 5 and 3 are eight and 1 are nine, he should say eight, nine, fourteen, seventeen, twenty. Then, in the column of tens, ten, fifteen, seventeen, twenty-five, twenty-six, thirty-two, thirty-three. This is called reading the columns. Let the pupils be often practised in it, both separately, and in concert in classes.

23. Add 8635, 2194, 7421, 5063, 2196, and 1245 together.

24. Add 246034, 298765, 47321, 58653, 64218, 5376, 9821, and 340 together.

25. Add 27104, 32547, 10758, 6256, 704321, 730491, 2587316, and 2749104 together.

26. Add 1, 37, 39504, 6890312, 18757421, and 265 together.

27. What is the sum of the following numbers, viz: seventy-five; one thousand and ninety-five; six thousand four hundred and thirty-five; two hundred and sixty-seven

thousand; one thousand four hundred and fifty-five; twentyseven millions and eighteen; two hundred and seventy millions and twenty-seven thousand?

28. What is the sum of 372856, 404932, 2704793, 9078961, 304165, 207708, 41274, 375, 271, 34, and 6?

29. What is the sum of 4073678, 4084162, 3714567, 27413121, 27049, 87419, 27413, 604, 37, and 9?

30. What is the sum of 36704321, 2947603, 999987, 76, 47213694, 21612090, 8746, 31210496, and 3021?

31. Add together fifty-eight billions, nine hundred and eighty-two millions, four hundred and eighty-seven thousands, six hundred and fifty-four; seven hundred and forty billions, three hundred and fifty millions, five hundred and forty thousands, seven hundred and sixty; four hundred and twenty-five billions, seven hundred and three millions, four hundred and two thousands, six hundred and three; thirtyfour billions, twenty millions, forty thousands and twenty; five hundred and sixty billions, eight hundred millions, seven hundred thousands and five hundred.

APPLICATIONS.

35. In all the applications of arithmetic, the numbers ad ded together must have the same unit.

In the question, How many head of live stock in a field, there being 6 cows, 2 oxen, 3 steers, and 15 sheep, the unit is 1 head of live stock. And the same principle is applicable to all similar questions.

QUESTIONS FOR PRACTICE.

1. How many days are there in the twelve calendar months? January has 31, February 28, March 31, April 30, May 31, June 30, July 31, August 31, September 30, October 31, November 30, and December 31.

Ans.

2. What is the total weight of seven casks of merchandise; No. 1, weighing 960 pounds, No. 2, 725 pounds, No. 3, 830 pounds, No. 4, 798 pounds, No. 5, 698 pounds, No. 6, 569 pounds, No. 7, 987 pounds?

3. At the Custom House, on the 1st day of June, there were entered 1800 yards of linen; on the 10th, 2500 yards; on the 25th, 600 yards; on the day following, 7500 yards; and the last three days of the month, 1325 yards each day: what was the whole amount entered during the month?

Ans.

4. A farmer has his live-stock distributed in the following manner: in pasture No. 1, there are 5 horses, 14 cows, 8 oxen, and 6 colts; in pasture No. 2, 3 horses, 4 colts, 6 cows, 20 calves, and 12 head of young cattle; in pasture No. 3, 320 sheep, 16 calves, two colts, and 5 head of young cattle. How much live-stock had he of each kind, and how many head had he altogether?

Ans.

horses, cows, oxen, colts, calves, head of young cattle, and sheep. Total live-stock, head.

5. What is the interval of time between an event which happened 125 years ago, and one that will happen 267 years hence?

6. There are 60 seconds in a minute, 3600 in an hour,

35. What principles govern all the additions in Arithmetic? What is the unit in the question? How many head of cattle in a pasture?