7. How many square yards in a figure 3 feet long and 3 feet wide? 6 feet square? 10 feet long and 9 feet wide? 6 feet long and 2 feet wide? (2 x 6=12÷÷9=1} yds., Ans.) In a figure 10 feet long and 4 wide? A. 4 yds.

8. How many square yards in 9 square feet? In 108? In 72? In 99? In 27? In 80? In 37?

Q. How, then, must square feet, square inches, &c. be divided? A. Square inches by square inches, square feet by square feet, &c.

Q. We are now prepared to answer that interesting question which occurs in geography, viz. the difference between miles square and square miles. The figures on the right are introduced for the purpose of its illustration. Examine them attentively, then tell me, for instance, What is the difference between 5 square miles and 5

miles square? A. 5 square miles means 5 miles in length and only 1 in breadth; but 5 miles square means 5 miles in length and 5 miles in breadth, making 5 times as many miles as only 1 in. breadth; that is, 25 square miles.

From these illustrations we derive the following general

RULE.

I. How do you proceed to find the contents of a square or parallelogram? A. Multiply the length by the breadth.

Exercises for the Slate.

1. In a room 16 feet long and 11 feet wide, how many square feet? A. 176.

2. How many acres in a piece of land 560 rods long and 32 rods wide? 560 X 32=112 square acres, Ans.

The pupil must recollect that square inches must be divided by square inches, square yards by square yards, &c.

3. How many acres in a piece of land 370 rods wide and 426 rods long? A. 985 acres, 20 rods.

4. How many rods long must a piece of land be, which is 80 rods wide, to make 1 acre? (2) How many rods wide to make 4 acres? (8) How many rods wide to make 200 acres? (400) .9. 410 rods,

5. How many square feet of boards are contained in the floor of a room 40 ft. 6 in. long and 10 ft. 3 in. wide? (Reduce the inches to the decimal of a foot.) A. 415,125 ft.=415} feet.

6. How many acres are contained in the road from Boston to Providence, allowing the distance to be 40 miles, and the average width of the road 4 rods? A. 320 acres.

7. How many square feet are contained in a board 12 inches long and 12 inches wide? (1) 12 inches wide and 24 inches long? (2) 3 feet long? (3) 20 feet long? (20) A. 26 feet.

8. How many square feet in a board 1 ft. 6 in. wide and 18 ft. 9 in. long? A. 28,125 ft. 281 feet.

9. How many yards of carpeting, that is 14 yd. wide, will cover a floor 21 ft. 3 in. long and 13 ft. 6 in. wide?

A. 25 yards. 10. How many feet of boards will it take to cover the walls of a house 30 ft. 6 in. wide, 40 ft. 9 in. long, and 20 ft. high? and what will they come to at $10 per 1000 feet? A. 2850 feet; cost $28.

11. How many shingles will it take to cover the roof of a barn 40 feet long, allowing the length of the rafters to be 16 ft. 6 in., and 6 shingles to cover 1 square foot? what will they cost, at $1,25 per 1000? A. 7920 shingles; cost $9,90.

12. What will a lot of land, 300 rods wide and 600 rods long come to, at $15 an acre? A. $16875.

13. What will a lot of land, 1 mile square, come to, at $20,75 per acre? A. $13280.

↑ LXXX. SOLID, OR CUBIC MEASURE.

Q. When a block is 1 inch long, 1 inch thick, and 1 inch wide, how many solid inches is it said to contain? A. 1 solid or cubic inch.

Q. How many solid feet does a block, that is 1 foot long, 1 foot thick, and 1 foot wide, contain? A. 1 solid or cubic foot.

Q. If a block 1 foot thick, 1 foot wide, and 1 foot long, contains 1 solid foot, how many solid feet does such a block that is 2 feet long contain? 3 feet long? 5 feet long? 10 feet long? 20 feet long? 30 feet long?

Q. How many solid feet does a block 2 feet long, feet thick, and 1 foot wide, contain? 2 feet wide? 3 feet wide?

Q. How many solid inches does a block 3 inches long, 2 inches wide, and 1 inch thick, contain? 2 inches thick? 4 inches thick? 10 inches thick?

Q. How, then, would you proceed to find how many solid feet, inches, &c. are contained in a solid body? A. Multiply the length, breadth and depth together.

1. How many solid feet in a block 4 feet thick, 2 feet wide, and 5 feet long? Ans. 4 X 2 X 540 solid feet.

2. How many solid or cubic feet in a block 12 inches long, 12 inches wide, and 12 inches thick? A. 1 solid foot.

Q. When a load of wood contains 128 solid feet, what is it called? A. 1 cord.

3. How many solid feet in a pile of wood 8 feet long, 4 feet wide, and 4 feet high? A. 1281 cord. How many cords of wood in a pile 8 feet long, 4 feet wide, and 8 feet high?

A. 256 solid feet: =

=2 cords. Q. In common language, we say of load of wood brought to market, if it is & feet long, 4 feet high, and 4 feet wide, that it is a cord, or it contains 8 feet of wood. But this would make 128 solid fect; what, then, is to be understood by saying of such a load of wood, that it contains 8 feet of wood? or, in common language, "there is 8 fect of it."

A. As 16 solid feet, in any form, are of 128 feet, that is, § of a cord, it was found convenient, in reckoning, to call every 16 solid feet 1 cord foot; then, 8 such cord feet will make 128 solid feet, or 1 cord, for 8 times 16 are 128.

Q. How, then, would you bring solid feet into cord feet? A Divide by 16.

4. How many cord feet in a pile of wood 8 feet long, 2 feet high, and 1 foot wide? How many in a load 8 feet long, 2 feet high, and 2 feet wide? 8 feet long, 4 feet wide, and 2 feet high?

5. If, in purchasing a load of wood, the seller should say that it contains 3 cord feet, how many solid feet must there be in the load? How many solid feet to contain 4 cord feet? 5 cord feet? 6 cord feet? 7 cord feet? 8 cord feet? 9 cord feet?

6. How many cord feet in a pile of wood 8 feet long, 1 foot wide, and 4 feet high?

In performing this last example, we multiply 4 feet (the height) by 1 foot (the width), making 4; then, this 4 by 8 feet (the length), making 32÷ 16 (cord feet), 2 cord feet, Ans. But, instead of multiplying the 4 By the 8 feet in length, and dividing by 16, we may simply divide by 2, without multiplying; for the divisor, 16, is 2 times as large as the multiplier, 8; consequently, it will produce the same result as before, thus: 4X14÷2=2 cord feet, Ans., as before.

Q. When, then, a load of wood is 8 feet long, or contains two lengths, each 4 feet (which is the usual length of wood prepared for market,) what easy method is there of finding how many cord feet such a load contains? A. Multiply the height and breadth together, and divide the product by 2.

7. How much wood in a load 8 feet long, 3 feet high, and 2 feet wide? 3×2=6÷2=3 cord feet, Ans.

8. How many cord feet in a load of ood 2 feet high, 2 feet wide, and of the usual length? 3 feet high and 2 feet wide ? 3 feet wide and 3 feet high? 4 feet wide and 4 feet high? 4 feet wide and 6 feet high? How many cords in a load 4 feet high, 4 feet wide?

9. How wide must a load of wood be, which is 8 feet long and 1 foot high, to make 1 cord foot? How wide to make 2 co, feet? 3 cord feet? 6 cord feet? 10 cord feet?

10. What will a load of wood 8 feet long, 3 feet wide, and 4 feet high, cost, at $1 per foot?

The foregoing remarks and illustrations may now be embraced in the following

RULES.

I. How do you find the contents of any solid or cube? A. Multiply the length, breadth and depth together.

II. When the length of wood is 8 feet, how can you find the number of cord feet it contains, without multiplying by 8 and dividing by 16? A. Multiply the breadth and height together, and divide the product by 2; the quotient will be cord feet. III. How do you bring cord feet into cords? A. Divide by 8.

Note. If the wood is only 4 feet in length, proceed as last directed; then, as 8 feet in length is 2 times as much wood as only 4 feet in length, hence the result found, as above, will be the answer in cord feet; that is, divide by 2 twice, or once by 4.

Exercises for the Slate.

1. How many solid feet in a load of wood 8 feet long, 4 feet wide, and 34 feet high? 4X 314÷27 cord feet, Ans. 2. How many feet in a load of wood 5 ft. 6 in. high, 3 ft. 9 in. wide, and of the usual length?

(Reduce the inches to the decimal of a foot. A. 10-3125=10% ft. Perform this last example by reducing the inches of a foot to a common fraction. This method, in most cases, will be found preferable: thus, taking the last example:

5 ft. 6 in. = 54 ft. =; then, 3 ft. 9 in. =34 ft. = 4 × 1 - 185 ÷ 2 = 182 = 10%, Ans., as before.

=

3. In a block 8 ft. Gin. in length, 3 ft. 3 in. wide, and 2 ft. 9 in. thick, how many solid feet? A. Decimally 75,96875 feet 75 feet. By common fractions; 1×13 × 2421 75 feet, Ans., as before.

4. If a load of wood is 8 feet long and 3 feet wide, how high must it be to make 1 cord?

In this example, we know that the height multiplied by the width, and this product divided by 2, must make 8 cord feet, that is, 1 cord or load; hence, 8X2=1635 feet, height, Ans.

5. If a load of wood is 5 feet high, and 8 feet long, how wide must it be to make 2 cords?

2 cords 16 cord feet; then, 16×2=32 ÷ 5} 6 feet wide, Ans. 6. If a load of wood is 5 feet high and 8 feet long, how wide must it be to make 3 cords? (9) 4 cords? (12) 8 cords? (24) A. 45 feet.

7. How many solid feet of timber in a stick 8 feet long, 10 inches thick, and 6 inches wide? (3) 10 feet long, 12 inches thick, and 1 ft. 3 in. wide? (123) 20 ft. 6 in. long, 24 inches wide, and 1 ft. 9 in. thick? (713) A. 8775 ft.

8. In a pile of wood 10 feet wide, 3 ft. 3 in. high, and 1 mile long, how many cord feet, and how many cords? A. 10725 cord feet

1340g cords. 9. How many tons of timber in 2 sticks, each 30 feet long, 20 inches wide, and 12 inches thick? A. 100 feet÷502 tons. 10. How many bricks 8 inches long, 4 inches wide, and 24 inches thick, will build a wall in front of a garden, which is to be 240 feet long, 6 feet high, and 1 foot 6 inches wide? A. 51840 bricks.

rived?

DUODECIMALS.

XXXI. Q. From what is the word duodecimals deA. From the Latin word duodecim, signifying twelve. Q. In common decimals, we are accustomed to suppose any whole thing, as a foot, for instance, to be divided into ten equal parts; but how is a foot dividea in duodecimals? and what are the parts called? A. Into twelve equal parts, called inches or primes, and each of these parts into twelve other equal parts, called seconds; also each second into twelve equal parts, called thirds, and each third into twelve equal parts, called fourths, and so on to any extent whatevers Q. What, then, are duodecimals? A. Fractions of a foot. Q. What fraction of a foot is 1 inch? A. Q. What fraction of a foot is 1 second? Q. What fraction of a foot is 1 third? A. 1 Q. What fraction of a foot is 1 fourth?

A.

ft.

of th

of 1 of 1 =1728 ft.

A. 12 of 11⁄2 of th of 15 = 20736 ft; Q. Now, since 12ths multiplied by 12ths make 144ths, and 12 make, make 4, it is

also, 144ths multiplied by 12ths make 1728ths, and plain that we may write the fractions without their denominators, by making some mark to distinguish them. What marks are generally used for this purpose? A. 12ths, inches, or primes, are distinguished by an accent, thus; 8′ signifies 2,8 inches, or 8 primes; 7" = TẪ4, or 7 seconds; 6""=T728, or 6 thirds, &c.

Q. We have seen that 12ths multiplied by 12ths produce 144ths; what, then, is the product of 5' (inches or primes) multiplied by 7' (inches)? A. 35", that

Q. What is the product of 5" (seconds) multiplied by 7' (inches)? A. 35''', that is, 35 thirds.

Q. What is the product of 5" (seconds) multiplied by 7" (seconds)?

A. 35, that is, 35 fourths.