4. By the census of 1850 the entire population of the United States was 23191876, and by the census of 1860 it was 31224885: what was the increase in 10 years?

5. In 1862 there were 10869 miles of railroad in Great Britain, and 33222 miles in the United States: how many more miles in the United States than in Great Britain?

6. An army of 30340 men lost 7568 in battle: how many men did it then contain?

7. In 1862 Ohio produced 35442858 pounds of butter and 20637235 pounds of cheese: how much more butter than cheese was produced?

8. A merchant having $11315 in bank, drew out $978: how much remained in the bank?

DEFINITIONS, PRINCIPLES, AND RULE.

Art. 29. Subtraction is the process of finding the difference between two numbers.

The Difference or Remainder is the number found by subtracting one number from another.

The Minuend is the number diminished.

The Subtrahend is the number subtracted.

Art. 30. Only Like Numbers can be subtracted. We can not take 3 pencils from 7 books, nor 3 units from 7 tens.

It is

Art. 31. The Sign of Subtraction is read minus or less. It shows that the number after it is to be subtracted from the number before it.

I. A.-4.

Art. 32. RULE FOR SUBTRACTION.-1. Write the subtrahend under the minuend, placing units under units, tens under tens, hundreds under hundreds, etc.

2. Begin at the right, and subtract each term of the subtrahend from the term above it, and write the difference underneath.

3. When any term of the subtrahend is greater than the term above it, add 10 to the upper term, and then subtract, and write the difference as before.

4. When ten has been added to the upper term, add 1 to the next higher term of the subtrahend before subtracting.

PROOF. Add the remainder and subtrahend. If their sum is equal to the minuend, the work is correct.

NOTE.-Instead of adding 1 to the next term of the subtrahend, 1 may be subtracted from the next term of the minuend.

LESSON VI.

Problems combining Addition and Subtraction.

1. Robert picked 21 peaches, and gave 7 to his sister and 8 to his brother: how many peaches had he left?

2. A garden contains 17 pear trees, 8 plum trees, and a certain number of peach trees: if there are 33 trees in the garden, what is the number of peach trees?

3. A grocer bought 35 bushels of apples, and

sold 17 bushels to A, 9 bushels to B, and the rest to C: how many bushels did he sell to C?

4. Jane is 8 years old and Lucy 13, and the sum of Jane's and Lucy's ages, less 7 years, is the age of Mary: how old is Mary?

5. A man bought a firkin of butter for $17, a crock of lard for $8, and a barrel of flour for $9; but he had not money enough by $7 to pay for them how much money had he?

6. A man earned $45, and paid $15 for house rent, $8 for flour, $7 for shoes, and $10 for groceries: how much had he left?

7. A man sees 15 pigeons on one branch of a tree, and 9 pigeons on another branch: if 7 should fly away, how many would be left on the tree?

8. A farmer had 23 chickens, but 7 of them were stolen and 5 were carried off by a hawk: how many chickens had he left?

9. A drover bought 17 sheep of one farmer, 9 sheep of another, and 8 of another, and then sold 7 of them to a butcher: how many sheep had he left?

10. A man gave a watch and $9 in money for a horse valued at $75: what did he get for his watch?

WRITTEN EXERCISES.

1. From a piece of carpeting containing 150 yards, a merchant sold 3 carpets, containing respectively 27, 39, and 42 yards: how many yards were left?

2. Rhode Island contains an area of 1306 square miles; Delaware, 2120; Connecticut, 4674; New Jersey, 8320; Maryland, 9356; and New York, 47000: how many more square miles has New York than the other five States named?

3. A regiment entered the service with 1088 men: 150 were killed in battle, 65 died from disease, 24 deserted, and 250 were discharged: how many remained?

4. A grain dealer bought 1250 bushels of wheat on Monday, 2145 bushels on Tuesday, and 3240 bushels on Wednesday, and on Thursday, fearing a decline in price, he sold 5450 bushels: how much wheat had he left?

5. A man deposited $175, $141, $75, $304, and $250 in a bank, and then drew out $480 and $225: how many dollars remained in the bank?

6. A railroad train left Cincinnati for St. Louis with 336 passengers, and during the trip 145 passengers came aboard, and 208 passengers left: how many were in the cars when the train reached St. Louis?

7. In 1860 the population of Maine was 628279; New Hampshire, 326073; 326073; Vermont, 315098; Massachusetts, 1231066; Connecticut, 460147; Rhode Island, 174620; and New York, 3880735. How many more inhabitants in New York than in the six New England States?

8. A man gave to his eldest son $2380; to the second, $245 less than to the eldest; and to

the youngest, $450 less than to the second: how much did he give to all?

9. From the sum of 2348 and 1864 subtract their difference.

10. From the sum of 506703 and 340067 take. their difference.

SECTION IV.

MULTIPLICATION.

LESSON I.

Multiplicand Figures, 1, 2, and 3.

1. Twice 2 are how many? 4 times 2? 6 times 2? 5 times 2? 8 times 2? 9 times 2?

2. Twice 3 are how many? 3 times 3? 5 times 3? 4 times 3? 7 times 3? 9 times 3?

3. How many are 5 times 1? 5 times 3? 7 times 1? 7 times 2? 8 times 1? 8 times 3?

4. A boy has 2 hands: how many hands have 6 boys? 8 boys? 10 boys?

5. There are 3 feet in a yard: how many feet in 2 yards? 4 yards? 6 yards?

6. If 3 bushels of apples fill a barrel, how many bushels will fill 3 barrels ? 5 barrels? 7

barrels ?

7. If a man earn 3 dollars a day, how many dollars will he earn in 6 days?