69. Because it is with a unit that we begin to form numbers, and proceed towards the left, in adding large numbers, we must begin by adding the unit figures, then the tens, and

so on.

70. Also, because ten units of any order are expressed by one unit of the next order on the left, we carry one to the next order for every ten which we find in the sum of any order

of units.

71. When the sum of any order is an exact number of tens; that is, when its unit figure is 0, this must be written underneath the order, and the tens-which (49) comprise all the other figures-must be carried to the next order. If we omit this cipher, the digits on the left will (56) be ten times less than they should be.

Examples.

1. 45395776+811405.

We begin on the right, and first add the unit figures, saying: 1 and 6 is 7, and 5 is 12 and 3 is 15. Then, taking room on the right of the sign as 15 is ten and 5, we write 5, and carrying 1, which is ten, to the place of tens, we add the figures in that place, saying: 1 and 8 is 9; and 7 is 16; and 9 is 25; and 5 is 30. Now, as this is 30 tens, or 3 hundreds, we place 0 on the left of 5; and carrying 3, we add the hundreds, saying: 3 and 7 is 10, and 4 is 14; which we write on the left of 05, and have 1405 for the sum. The figure 1, which is ten hundreds, being, in accordance with the general law, in the units place of thousands.

To prove the work, we begin on the left, and add thus: 3 and 5 is 8; and 6 is 14; and 1 is 15. We write 5, saying 5 and go 1 to 5 is 6; and 9 is 15; and 7 is 22; and 8 is 30; ◊ and go 3 to 4 is 7; and 7 is 14.

Let the following numbers be added in the position in which they stand, and the work proved as above:

4. 51573 +358 + 9256 + 19 =

5. 92+1509 +435 + 79847 +5840=

6. 317258040 + 92273 +38029

72. When the numbers to be added are not only large, but numerous, it is usual, for the sake of convenience, to place them under each other, so that units of the same order may stand in the same column. This is easily done, in placing

the unit figures under each other, and the other figures in regular order towards the left; whereby, as all take their value by the same general law, from their position, as they advance in this direction, each figure will be found in a column with figures of the same order of units-that is, if units be under units, tens will be under tens, hundreds under hundreds, &c. We then begin with the units, and add as usual; taking care, after writing the unit figure of the sum of each column, to write also, underneath this figure, the tens which are to be carried to the next column. This is done for the sake of reference, and is very useful in long additions; for if, while adding, any thing should call off the attention, we are not obliged to resume the work from the beginning.

In writing the numbers of the following examples, to facilitate the work, we first write under each other those of the lowest order; then those of the next higher, and so on till all are written.

Example.

To add the numbers four hundred and ninety-six; three thousand seven hundred and forty-nine; nine hundred and thirty-seven; seventeen thousand and eighty-five; five millions nine hundred and eighty-eight thousand four hundred and seventy-three; we place them thus:

M. Th. U.

496

937

3749

17085

5988473

6010740
112233

Then, beginning at the right-hand column, (for the sake of greater despatch, we omit all intermediate language,) and say: three, eight, seventeen, twenty-four, thirty. We write 0 under the column of units. Under this 0 we also write the 3 tens, which we carry to the next column, and say: ten, eighteen, twenty-two, twenty-five, thirty-four. Now, as this is 34 tens, or 3 hundreds and 4 tens, we write 4 under tens, and under this 4 the 3 hundreds, which we carry to the next column. In the same manner we proceed with the other columns till the operation is finished.

To prove the work, we begin at the top of each column and add downwards, taking them in order from right to left.

73. The scholar, having acquired some facility in adding, should entirely drop the use of intermediate language, except in carrying from one column to another, as in the following example:

6935

4872

5888

5975

7978

31648

432

where we say simply, 8, 13, 23, 28; 8 and go 2 to 7 is 9, 16, 24, 34; 4 and go 3 to 9 is 12, 21, 29, 37, 46; 6 and go 4 to 7 is 11, 21, 31, which last we set down in full.

3. Add together three hundred and thirty-six; ninetyseven; three thousand and seventy-four; eight hundred and eighty-seven; and six hundred and forty-three thousand eight hundred and nine.

Answer. Six hundred and forty-eight thousand two hundred and three.

4. Required the sum of eighty-two; four hundred and seventeen; forty-nine; three thousand seven hundred and thirty-eight; and eleven millions five hundred thousand and twenty-four.

Ans. Eleven millions five hundred and four thousand three hundred and ten.

5. Required the sum of seven thousand two hundred and nineteen; sixty-three; five hundred and sixty-four; fourteen millions seven hundred and nine thousand and forty-five; twenty-seven thousand nine hundred and ninety; seventyseven; nine thousand and ninety-seven; and ninety millions thirty-three thousand six hundred.

Ans. One hundred and four millions seven hundred and eighty-seven thousand six hundred and fifty-five.

6. Required the sum of two thousand three hundred and fifty-six; sixty-six; three hundred and nineteen; twelve millions five hundred and seven thousand and twenty-six; ninetyseven hundred thousand and fifty-nine; four thousand

four;

six hundred and seventy-four; and nine hundred and eight millions and six.

Ans. Nine hundred and twenty-one millions two hundred and fourteen thousand six hundred.

7. Required the sum of thirty-seven; six hundred and fifty-five; three thousand and seventy; twenty millions ninetyfour thousand and fifty-seven; seventy-four thousand and fortyseven; six thousand and ninety-nine; three hundred and twenty-eight; fifty-seven thousand five hundred and six; and six hundred and six millions five hundred thousand seven hundred.

Ans. Six hundred and twenty-six millions seven hundred and thirty-six thousand four hundred and ninety-nine.

8. What is the sum of fifty-eight; three thousand and four"; eighty-three; ten thousand four hundred and ninety; three hundred and fourteen; six millions six hundred and four thousand and seventy-five; two hundred and twenty-four; eleven thousand and fifty; and five billions one hundred and four thousand and seventy-two.

Ans. Five billions six millions seven hundred and thirtythree thousand three hundred and seventy.

Let the following examples be performed and proved, as

above:

74. The unit figure of the sum of any column, being of the order of the column, is written underneath; and as the other figures of this sum express (48) a number of tens with regard to that figure, they are carried, as that number, to the next column on the left; the figures of which, (36) in the same sense, are also tens. Therefore, when the sum of a column equals or exceeds 100, the number to be carried will be 10 or more: when the sum equals or exceeds 200, the number to be carried will be 20 or more: when the sum equals or exceeds 300, the number to be carried will be 30 or more, &c. On reaching 100, the scholar may make a dash at the side of his work, and proceed with the surplus, as at first.