2. What are the complements of 2222 and of 73480.

3. Why does the first 2 on the right of No. 1, require 8 as its complement, while all the others require only 7? [Add the complement and see.]

No. 1. Subtract'n by adding complement. No. 2. Subtract'n in usual way.

756454 Compl. of Subt.

Dropping 1,-582946 True remainder.

It is proper to remark here, that in the example " No. 1. Subtraction by adding complement," the third and fourth lines are altogether superfluous, being placed there merely to exemplify. All that is necessary in such operations is to add (without having it written) the complement of the subtrahend to the minuend, dropping 1 of the next higher rank of figures than the highest of the subtrahend.

4. What is the minuend? The subtrahend? The difference, or remainder? By how much is the minuend greater than the remainder? Ans. By the sIf, in subtrac

tion, then, you announce the minuend as the answer in place of the remainder, as in the above example No. 1, how much too large will your answer be? [See answer to last question but one.] Now, as your answer is too much by the amount of the subtrahend, how much too large will it be if you add to it the complement of the subtrahend; that is, how much does a number and its complement amount to? If the answer, then, be too great by the sum of the subtrahend and its complement, how can it be rectified? Ans. By dropping 1 of the

May not, then, the following be considered as the eighth principle of arithmetic ?

VIII. The difference of two numbers may be obtained by adding to the larger the complement of the smaller, and diminishing this sum by 1 of the next higher rank of figures than the highest of the smaller.

5. Read off the difference of each of the following pairs of numbers, by adding, without writing down or naming, the complement of the smaller to the larger, and diminishing the amount by 1 of the next higher rank of figures than is contained in the smaller, and repeat similar exercises on the black-board and slate till it can be done rapidly.

4 16 5 15 6 17 3 5 7 6 17 15 18 7 6 4 9 9 8 8 9 1 14 3 18 2 9 9 18-7 25-4 7-2 8-4

8-2 17-3 14-5 9-0

[All the exercises in subtraction given above may now be performed, by adding the complement, in place of taking the subtrahend from the minuend.]

[On the same principle, the sum of two or more numbers may be taken from a minuend, whether it consist of one or more numbers, without finding their sum or sums, as in the following examples :]

6. From 46589 take the sum of 2976, 3582, 176, and 24, by addition.

Proof

39831 Difference.

Sum of difference and subtrahend.

Solution. - 6, 4, 8, 4 (comps.) and 9-31; carrying 3 to 7, 2, 1, 2 (comps.), 8=23; dropping 1 gives 1 to 8, 4, 0 (comps.), 5-18; dropping one, we have 6, 7 (comps.), 6=19; dropping 2 (why 2?) from 1 and 4=3.

7. From 7962, take the sum of 5143, 236, 728, 97, 4, and 8, by addition, and prove by subtraction in the usual manner. 8. From 549728, take the sum of 72, 3146, 458, 6, 93, and 872, by addition, and prove by subtraction.

9. From 82493 take 725, 4193, and 6127, and prove by subtraction.

10. From 7248'63 take the sum of 24'5, 784-26, 3158, and 234, and prove by subtraction.

11. From 946 783 take the sum of 71.375, 42'6, and 84'07, and prove.

12. From 8148 take the sum of 7'05, 3'56, 92'4, and 145'3, and prove.

13. Find the difference between the sum of 7643 and 5234, and the sum of 6431 and 978, without finding those sums.

The process reads thus: 2+9 (comps. of 8 and 1) +4+3 18; carry 1 to 2+6 (comps. of 7 and 3)+3+4=16; . carry 1 to 0+5 (comps. of 9 and 4) +2+6=14; set down the 4 and drop the 1 (Why?); 3 (comp. of 6) +5+7=15; drop the 1.

14. From the sum of 5682 and 39476 take the sum of 2158 and 3426, and prove. [Here the 2 ones will be dropped from the rank of tens of thousands.]

15. From the sum of 3678, 5237, 4286, take the sum of 12 and 5213. [Here one will be dropped from the rank of hundreds, and one from that of tens of thousands.] Prove by subtraction.

16. From the sum of 276, 3854, 913'2, take the sum of 346, 271, and 1234. How many ones must be dropped in this operation, and from what ranks of figures? Prove by subtraction.

17. From 9876 take the sum of '71, 360'8, 215, 42'76. How many ones must be dropped in this operation, and from what ranks of figures? Prove.

18. From the sum of 8721, 345, and 26'38, take 145. How many ones must be dropped, and whence?

19. From the sum of 526, 3927, and 44, take the sum of 1234 and 600.

When subtraction is performed by addition of the complements, it should be proved by subtraction performed in the usual way. Exercises of this sort afford admirable means for the development of the judgment and imagination, as well as of the memory. The judgment is employed in deciding what number is to be dropped, and when; and the imagination is exercised in calling to mind one figure by the sight of another. Additional exercises may be given by the teacher when necessary; that is, when the pupil or class has not acquired a facility by those given above. It would be still better, however, if the pupil were accustomed to form them for himself.

Questions by the teacher.-What is subtraction? See p. 56, 3. What is the greater number called? The smaller? The result? What is the sign of subtraction? Its name? What is the most convenient mode of arranging the figures? How many modes should be used? Name them. Should subtraction be commenced at the right or left? Why? In what case is it of no consequence where it is commenced ? * Will the difference be changed if the same number be added to the minuend and subtrahend? How, then, should we proceed when a figure in the subtrahend is greater than the one of corresponding rank in the minuend? If the subtrahend and difference be given, how may the minuend be found? What, then, is the first mode of proving subtraction? When the minuend and difference are given, how may the subtrahend be found? What, then, is the second mode of proving subtraction? How may the sum of two or more given numbers be subtracted from another number, or from the sum of two or more given numbers, at one operation, without finding either of the sums? Addition and subtraction may be performed simultaneously by a much easier method than the above, but it does not afford such excellent mental discipline. As, however, it may be preferred for practical business, it is proper to present it here, as follows:

Find the difference between the sum of 2556 and 3798, and the sum of 1324 and 2796, by writing the complement of the subtrahend, and affixing to each complement a hyphen (-), to

*We may commence at the left in every case, provided we take notice as we proceed whether the adjoining figures on the right require 10 to be added, and act accordingly. Exercises so performed would afford excellent intellectual discipline.

show that one of the next higher denomination is to be omitted for each number in the subtrahend.

S 2556
3798

Minuend, {

Subtrahend,

-8676 Complement of 1324.
-7204 Complement of 2796.

Difference, 2234

Perform all the above exercises from 1 to 19, writing the complements of the numbers to be subtracted in place of the actual numbers.

Practical Exercises.

1. Washington died in 1799, at the what year was he born?

age

of 67 years. In

2. There are two adjoining farms, one of which was sold for $5820, and the other for $376 less. What was the cost of both? [This, and each of the five following exercises, should be performed at one operation by means of the complement.] Ans. $11,264.

3. A lady went a shopping with $24 in her purse. She paid $6 for a bonnet, $3 for two pairs of shoes, $5 for a piece of sheeting, and $3 for marketing. How much had she left? Ans. $7.

4. A farmer, who was in the habit of settling annually with his creditors, set out for that purpose on New Year's day with $150 in his pocket. He paid his blacksmith $20, his tailor $28, his shoemaker $25, his saddler $30, and his storekeeper $43. After making these payments, how much had he left? Ans. $4.

5. A farm, including the stock of cattle, sheep, horses, and hogs, was valued at $8000. The cattle were considered to be worth $240, the sheep $175, the horses $150, and the hogs $75. What was the value of the land? Ans. $7360.

6. A merchant sent his clerk to collect some accounts, and directed him to take a purse of silver with him, in case change should be wanted. The clerk collected from one person $25, from another $140, from another $256, and from another $67.