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Geometry.

329. THE CIRCUMFERENCE OF A CIRCLE. 1. Cut a 3-inch circle from cardboard. By rolling it upon a foot rule, measure its circumference.

2. Measure the diameter of a bicycle wheel; then by rolling it upon the ground or upon the school-room floor, measure its circumference.

3. In a similar manner measure the diameters and the circumferences of other wheels until you are convinced that the circumference of a circle is a little more than times its diameter.

4. The circumference of a circle is nearly 31 times the diameter ; more accurately, it is 3.141592+ times the diameter.

NOTE. - It is a curious fact that the diameter of a circle being given in numbers it is impossible to express in numbers its exact circumference. The circumference being given in numbers it is impossible to express in numbers its exact diameter. In other words, the exact ratio of the circumference to the diameter is not expressible.

5. Find the approximate circumference of a 5-inch circle ; of a 7-inch circle ; of a 10-inch circle. *

6. Find the approximate diameter of a circle that is 6 ft. in circumference. *

7. The circumference of a 6-inch circle is how many times the circunıference of a 3-inch circle ?

8. The diameter of a circle whose circumference is 12 inches is what part of the diameter of a circle whose circumference is 24 inches ?

* In the solution of such problems as these the pupil may use, as the approxi. mate ratio of the circumference to the diameter, 3.14.

330. MISCELLANEOUS REVIEW.

1. Find the approximate circumference of a circle whose dianieter is 3.4 meters.

2. Find the approximate diameter of a circle whose circumference is 3.4 decimeters.

3. Find the approximate diameter in yards of a circular 1-mile race track; of a half-mile track.

4. Find the approximate diameter in meters of a circular 1-kilometer race track ; of a half-kilometer track.

5. A 28-inch bicycle wheel will make how many revolutions in running one mile?

6. A 70-centimeter bicycle wheel will make many evolutions in running 1 kilometer ? *

See table on page 226 and give approximate answers to the fol. lowing:

7. Forty meters are how many yards ? 8. Forty yards are how many meters ? 9. Forty kilometers are how many miles ? 10. Forty miles are how many kilometers ?

11. Forty ars are how many acres ? 12. Forty acres are how many ars?

13. Forty sters are how many cords?
14. Forty cords are how many sters?
15. Forty liters are how many quarts ?
16. Forty quarts are how many liters ?
17. Forty kilograms are how many pounds ?
18. Forty pounds are how many kilograms?

* The exact answer to such questions as this cannot be found: but the approxi. mation is practically correct.

DENOMINATE NUMBERS.

Linear Measure.

Note.-In part to provide for ready reference, and in part to give further application of the principles presented on the preceding pages, the remaining pages of this book are devoted chiefly to denominate numbers.

331. The English and United States standard unit of length is the Imperial yard arbitrarily fixed by Act of Parliament and afterward adopted in the United States. It is about $$1$$$ of the length of a pendulum that vibrates once a second at the level of the sea in the latitude of London. It is 3834 of a meter.

TABLE.

164 feet

12 inches (in.) = 1 foot (ft )
3 feet = 1 yard (yd.)
51 yards

1 rod (rd.)

: 1 rod.
320 rods

: 1 mile (mi.)
1760 yards =1 mile.
5280 feet

1 mile.
1 fathom (used in measuring the depth of the sea) = 6 feet.
1 knot (used in navigation)

= 1.15 + miles. 1 league (used in navigation)

= 3 knots. 1 hand (used in measuring the heights of horses) = 4 inches. 1 chain (used by civil engineers)

100 feet. 1 chain (used by land surveyors)

66 feet. 1 pace (used in measuring approximately)

of a rod. 1 barleycorn (used in grading length of shoes) of an inch. 1 furlong (a term nearly obsolete)

= of a mile.

Denominate Numbers -Linear Measure.

EXERCISE.

or

1. Mont Blanc is 15810 feet, or about miles high. 2. Mt. Everest is 29000 feet, or about

miles high. 3. Commodore Dewey opened fire on the enemy at a distance of 5000 yards, or about miles. 4. My horse, measured over the front feet, is 164 hands,

feet inches high. 5. The vessel seemed to be about three leagues, or miles distant.

6. On sounding, they found the depth of the water to be. 15 fathoms, or feet.

7. The cruiser made 20 knots or about miles, an hour.

8. The length of the lot was 36 paces, or about rods. 9. 10000 feet is nearly miles. 10. 15000 feet is nearly miles. 11. 1000 yards is about of a mile. 12. 100 feet is rods foot. 13. 200 feet is rods feet. 14. 300 feet is rods feet. 15. A kilometer is about rods. 16. A Civil Engineer's chain is rods foot.

PROBLEMS.

1. A seven-foot drive wheel of a locomotive makes how many revolutions to the mile ?

2. Which is the longest distance, 5 miles 319 rods 16 feet 6 inches, 5 miles 319 rods 5 yards 1 foot 6 inches, or 6 miles ?

3. Reduce 40 rd. 4 ft. 5 in. to inches.
4. Reduce 1100 inches to yards, feet, and inches.

Denominate Numbers-Surface Measure

332. The standard unit of surface measure is a square yard which is the equivalent of a 1-yard square.* This unit, like the square foot, square inch, square rod, and square mile, is derived from the corresponding unit of linear measure.

TABLE.

=1

144 square inches (sq. in.) = 1 square foot (sq. ft.),
9 square feet

square yard (sq. yd.).
304 square yards

1 square rod (sq. rd.). 2724 square feet

1 square rod. 160 square rods

= 1 acre' (A.).
4840 square yards

1 acre.
43560 square
feet

1 acre.
640 acres

= 1 square mile (sq. mi.).

EXERCISE.

1. Show by a drawing that there are 144 square inches in a 1-foot square.

2. Show by a drawing that there are 9 square feet in a 1-yard square.

3. Show by a drawing that there are 304 square yards in a 1-rod square.

4. Estimate the number of square yards of blackboard in the room ; the number of square feet of blackboard.

5. Estimate the number of square feet in the floor of the schoolroom ; the number of square yards.

6. Estimate the square yards of plastering on the walls of the schoolroom.

7. Estimate the number of square rods in the schoolhouse lot. Is the lot more or less than of an acre ?

* A surface that is 9 ft. long and I st. wide is a square yard though it is not itself a square. It is the equivalent of a l-yard square.

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