figures are cut so that their bases are very small, they may be considered to be pyramids and their volume can be found by the method used for

pyramids.

The sum of all these pyramids is the sphere.

The sum

of the bases of these pyramids is the surface of the sphere,

and the altitude is the radius of the sphere.

Therefore, the sum of the volumes of these pyramids is one third the product of the surface of the sphere by its radius.

This gives the volume of a sphere as 1 × 4 πr2 × r, which is given more briefly as r3; and since rd, .. r3 = }}

П

· ·. § πr3 = § π × } d3 = TM, d3 = .5236 d3.

6

10. Find the volume of a sphere whose radius is 20 in. 11. Find the volume of a sphere, the radius being 8 ft. 12. The surface of a sphere equals 1257 sq. in. Find its volume.

13. The surface of a hemispherical dome is 2513.5 sq. ft. Find its diameter.

14. The volumes of similar solids are to each other as the cubes of their corresponding dimensions. How many times as large as the earth is the sun? The diameter of the sun is nearly 888,000 mi., and the diameter of the earth is nearly 8000 mi.

15. How many times as large as the moon is the earth? The moon's diameter is 2200 mi., nearly.

16. How many times as large as the earth is Saturn? The diameter of Saturn is 73,000 mi.

17. How many times as large as the earth is Jupiter? The diameter of Jupiter is 88,000 mi., nearly.

MEASURE OF TEMPERATURE

A thermometer is an instrument for measuring heat. The principle of the thermometer is that substances expand with heat according to a natural law.

There are two different styles of thermometers in general use, -the Centigrade and the Fahrenheit. The Centigrade is used on the continent of Europe for all purposes and in this country for scientific purposes. The Fahrenheit is commonly used in America as applied to "weather." The Centigrade thermometer marks the melting point of ice 0°, and the boiling point of water 100°. The interval between these points is divided into 100 parts, or degrees, so that the change in the volume of the mercury between any two consecutive marks is of the change from 0° to 100°.

The Fahrenheit thermometer divides the space between the melting point of ice and the boiling point of water into 180°. It marks the melting point of ice 32°, and the boiling point of water 212°. 0° is therefore 32° below freezing.

Notation. 92° C. means 92 degrees on the Centigrade thermometer.

45° Fahr. means 45° on Fahrenheit's thermometer. +10° means 10° above zero.

(1) To change from degrees Fahrenheit to degrees Centigrade, subtract 32° and multiply the remainder by §.

(2) To change from degrees Centigrade to degrees Fahrenheit, multiply the number of degrees Centigrade by g and add 32 to the product.

Explanation of the rules:

(1) Suppose the temperature on a Fahrenheit thermometer is n degrees. Subtract 32° to get the number of degrees from 0°. A difference of 180° Fahrenheit a difference of 100° Centigrade. Therefore, a difference of 1° Fahrenheit a difference of 50 Centigrade. Therefore, a difference of (n-32°) Fahrenheit (n - 32°) Centigrade, which symbolizes the first of the above rules.

=

(2) A difference of n° C. = a difference of n° Fahrenheit. Hence, n° C. = (f n° + 32°) Fahr.

21. Give the following table in the Fahrenheit scale.

18.

25. Water attains its maximum density at 4° C. Express this temperature on Fahrenheit's scale.

SPECIFIC GRAVITY

If the weight of a substance is divided by the weight of an equal volume of water, the quotient is the specific gravity of the substance. Specific gravity is a statement of how many times as heavy as water a substance is. Thus, cast iron is 7.21 times as heavy as water, and hence its specific gravity is 7.21. Cork is about one fourth as heavy as water, and hence its specific gravity is 1.1 cu. ft. of water weighs 1000 oz.

Example. Find the weight of 28 cu. in. of mercury.

SOLUTION. 1728 cu. in. of water weigh 1000 oz.

28

1728

× 1000 oz. = 13.57 oz., weight of 28 cu. in. of water.

× 1000 oz.= weight of 28 cu. in. of mercury.

EXERCISE 163

1. Find the weight of 1 cu. ft. of steel; 1 cu. ft. of glass; 1 cu. ft. of clay.

2. Find the weight of 1 cubic inch of water. Find the weight of 1 gal. of water.

3. A cubic foot of marble weighs 2700 oz. specific gravity of marble.

4. A cubic foot of sea water weighs 642 lb. specific gravity of sea water.

Find the

Find the

5. A cubic foot of goat's milk weighs 65 lb. how many times as heavy as water goat's milk is.

Find

6. The mercury in the barometer exactly counterbalances the pressure of the atmosphere. If the barometer is 30 in. high, find the pressure of atmosphere upon every square inch of surface.

7. A swimming pool of fresh water is 25 ft. by 16 ft. by 5 ft. Find the weight of the water it contains.

8. A block of granite is 6 ft. by 4 ft. by 2 ft. thick. Find its weight in tons.

9. Find the weight of 1 cu. in. of gold; of 1 cu. in. of silver; of 1 cu. in. of platinum; of 1 cu. in. of lead. 10. A block of ice 3 ft. by 2 ft. by 1 ft. thick weighs how many pounds?

11. What is the weight in tons of 1 cu. yd. of clay?

12. Find the weight of the butter required to fill a box 16 in. by 9 in. by 8 in.

13. Find the weight of the air in a hall 27 ft. by 24 ft. by 15 ft. 6 in. 1 cu. ft. of air weighs .08073 lb.

14. A cubic foot of coal weighs 814 lb. Find the specific gravity of coal.