EXERCISES. I.

1. A piece of lead weighs 3 lbs. 12 oz., and a piece of oak weighs 2 lbs. 8 oz.: what ratio does the mass of the piece of lead bear to the mass of the piece of oak?

2. The specific gravity of gold being 19.2 and that of lead being 11-2, find the ratio which the quantity of matter in 5 cubic inches of lead bears to the quantity of matter in 7 cubic inches of gold.

3. In a certain state of the atmosphere 100 cubic inches of air weigh 31 grains; at a temperature the same as that of the air 30 cubic inches of mercury weigh 14.88 lbs.: find the number of cubic inches of air which contain as much matter as a cubic inch of mercury.

4. If 100 cubic inches of oxygen (under certain circumstances of pressure and temperature) weigh 35 grains, and 1 cubic inch of mercury weighs 49 lbs., how many cubic inches of oxygen would contain the same quantity of matter as 1 cubic inch of mercury?

5. An iceberg floats with of its volume immersed: find the specific gravity of ice.

6. Define Mass and Density. Of two bodies one has a volume of 5 cubic inches, the other of one-fifth of a cubic foot. In a balance the former weighs 15 oz., the latter 12.8 lbs.: what is the ratio of the mass of the first to that of the second, and what is the ratio of the density of the first to that of the second?

7. A rod of uniform cross section, 18 inches long, weighs 3 oz.; its specific gravity is 88: what fraction of a sq. in. is the area of its cross section? The weight of a cubic inch of water may be taken to equal 252 grains.

8. If 8 cubic inches of a liquid whose specific gravity is 125 are mixed with 12 cubic inches of another liquid whose specific gravity is 1.125, what is the specific gravity of the mixture if neither shrinking nor expansion occurs?

9. A piece of glass weighs 47 grams in air, 22 grams in water, and 25 8 grams in alcohol: find the specific gravity of the alcohol.

10. A specific gravity bottle full of water weighs 44 grams, and when some pieces of iron weighing 10 grams in air are introduced into the bottle, and the bottle again filled up with water, the combined weight is 527 grams: what is the specific gravity of the iron? 11. Define density. The specific gravity of brass referred to water is 8.2. Taking the weight of one cubic foot of water as 1,000 oz., find the density of brass in pounds to the cubic inch

12. If a cubic centimetre of water contained exactly a gram of matter, what would be the quantity of matter-estimated in grams -in a cubic foot of lead, a linear foot being 30-45 centimetres, and the specific gravity of lead being 11:445,

13. Define clearly what is meant by Specific Gravity. Is there any difference in specific gravity between 4 lbs. of iron and 2 lbs. of the same metal?

14. A solid weighs in vacuo 100 grains, in water 85 grains, and in another fluid 88 grains: what is the specific gravity of this fluid ?

15. A solid, soluble in water but not in alcohol, weighs 346 grains in air and 210 in alcohol: find the specific gravity of the solid, that of alcohol being 0.85.

16. A glass ball weighs 1000 grains; it weighs 630 grains in water and 650 grains in wine: find the specific gravity of the wine.

17. The glycerine barometer is found to stand at 329.2 inches when the mercurial barometer stands at 3061; given that the specific gravity of mercury is 13.569, find the specific gravity of glycerine.

18. The specific gravity of a substance is 7.5; a portion of it weighs 390 grains in water: what is its weight in air?

Summary.

Measurement of area is obtained directly from the measurement of length.

The unit of area is the area of a square, the edge of which is the unit length. As the unit of length in the British system is the yard, the British unit of area is the square yard. The smaller units—the square foot and the square inch-are, however, much more convenient, and are generally used.

The Metric unit of area is the square metre, the square centimetre, or the area of a square, the length of its edge being one centimetre, is generally used.

Prefixes deci-, centi-, and milli- are used to denote sub-multiples of the unit, and the prefixes Deka-, Hekto-, and Kilo- are used for multiples of the unit.

Unit of volume in the British system is the gallon, defined as the volume occupied by 10 lbs. of pure water at a temperature 62° F. A larger unit is the cubic foot. A cubic foot is taken to weigh 62.3 lbs., or approximately 6 gallons, equal to 62 lbs., or 1000 ounces.

A Pint of pure water weighs about 14 lbs. The sub-multiple (the cubic inch) is often used; a cubic inch of water weighs

A cubic yard contains 27 cubic feet.

The metric unit of volume is the litre, or the space occupied by a kilogram of water at 4° C.; a more convenient unit for many purposes is the cubic centimetre (c.c.). The mass of a c.c. of pure water at 4° C. is one gram.

The mass of a body denotes the quantity of matter it contains. Unit of mass is one pound (1 lb.) defined as the quantity of matter in a platinum cylinder deposited in the Standards Office.

Multiples of the unit are 1 cwt. = 112 lbs., and 1 ton=20 cwts., or 2240 lbs.

Sub-multiples are obtained by dividing one pound into 16 equal parts, each an ounce, or into 7000 parts called grains.

The metric unit of mass is the Kilogram. This is defined to be the mass of a platinum cylinder deposited in the French archives. The kilogram is very nearly 1 litre, or 1000 c.c. A more convenient unit is the mass of 1 c.c. of water, or 1 gram.

Weight. The weight of a body is the attractive force which the earth exerts on a body at or near its surface. The weight of a body denotes a quantity of force. The mass of a body denotes a quantity of matter.

Density of a body is the mass of unit volume. Relative density of a body is the ratio of its mass to that of an equal volume of a standard substance. The standard substance is pure water, and relative density is usually known as specific gravity.

Principle of Archimedes.-A body immersed in water loses weight equal to the weight of the liquid displaced, A floating body displaces a volume of water equal to its own weight.

CHAPTER III.

TIME AND ITS MEASUREMENT.

ANGULAR MEASUREMENT; DEGREES, RADIANS. GEOMETRICAL CONSTRUCTION.

Intervals of Time.—A measure of time may be derived from the apparent motion of the sun which appears to an observer to move through certain periodic changes of position; thus, it rises and travels higher and higher until it reaches its highest point, then descends lower and lower until it finally sets. It is difficult, if not altogether impossible, to observe the instant at which dawn commences or darkness begins. It is, however, easy to tell when the sun is at its highest point in the heavens, for the shadow cast by any upright object on the earth's surface would shorten as the sun in its apparent journey travelled higher and higher; at noon, when the sun reaches its highest position, the shadow would have its shortest length, and as the apparent journey continues, the shadow again lengthens.

An Apparent Solar Day is the interval of time reckoned from one noon to the next; true noon is the instant when the sun rises to its highest altitude, or, as it is often expressed, when the sun "souths." These apparent days are variable in length, being shortest in November, and longest in February.

Mean Solar Day.-The apparent solar day marks out unequal portions of time; a mean or average solar day is obtained by adding together the lengths of all the days in the year and dividing the sum by the number of days; this gives a mean or average interval of the same duration in each case.

As already indicated, the interval between two consecutive southings of the sun varies throughout the year, but this is

not the case with the southing or transit of a star; it is found that the time between two successive transits of a star is invariably the same.

Rotation of the Earth. The apparent rotations of the stars across the sky are produced by the rotation of the earth, and the interval between any two consecutive returns of a star to the same point of the sky is the exact time taken for one rotation of the earth on its own axis, and is called a sidereal day. On this uniformity our standards of time are made to depend. The result is the same whatever star is selected, thus showing that any and every part of the earth's surface moves with the same angular velocity.

The invariable interval of time between two consecutive southings of a star is divided into 24 equal parts, each called an hour. The hour is further subdivided into 60 minutes, and each minute into 60 seconds. Fractional parts of a second are expressed decimally.

A Mean Solar Year consists of 366 242 sidereal days, or 365 242 mean solar days.

Measurement of Time.-The measurement of time by means of clocks and chronometers can be made with great accuracy and precision. These are checked by astronomical observations, especially by means of the transits of stars. By means of one or more good clocks carefully made to agree with astronomical observations, any number of others can, by means of electrical appliances, be made to record accurately the same time as the standard clocks.

Unit of Time.-The unit of time chiefly used is the second. Engineers and others, in the estimation of power, etc., use the larger unit, the minute.

Angular Measurement. In angular measurement, as in linear measure, a suitable unit of measurement is selected, and the number of times that any given angle contains the unit is the numerical measure of the angle.

The two units in general use are the degree and the radian. The degree is obtained by drawing a circle of any convenient radius, and dividing its circumference into 360 equal parts. If two consecutive divisions be joined to the centre, the two lines so drawn contain a length of arc equal to goth part of the