13. Units of Time.-The scientific unit of time, both in the British and the metric system, is the mean solar second. The mean solar second is one 86,400th part of a mean solar day. The mean solar day is the average interval which elapses between successive transits of the sun across the meridian at any place during a whole year.

Owing to the excentricity of the earth's orbit, and the fact that the earth's axis is not perpendicular to the plane of the orbit, the interval between two successive transits varies during the year, so that the actual solar day is not the same as the mean solar day.

If a clock keeps mean time and agrees with solar time, that is time such as would be indicated by a sundial, when the sun appears in that portion of the heavens known as the first point of Aries, then the difference between the time of noon as indicated by this clock, and the time when the sun crosses the meridian on any day, is called the equation

of time at noon for that day. The curve in Fig. 1 gives the equation of time for the year. When the curve is above the axis Ox the equation of time is positive, that is, the time as shown by a mean-time clock will be ahead of the transit of the sun by the amount shown by the ordinate.

It will be seen that the equation of time is zero, that is, the time

as indicated by a mean-time clock and the sun will be the same on April 15, June 15, August 31, and December 24. On February 11 the mean-time clock is 14 minutes 29 seconds ahead of the sun, while on November 1 it is 16 minutes 20 seconds behind the sun.

The unit of time used in astronomy is the sidereal day. This represents the interval between two consecutive transits of one of the fixed stars across the meridian. Since the distance between the earth and any of the fixed stars is very great, compared even with the diameter of the earth's orbit, the line joining the earth to such a star remains always parallel to itself. Hence the sidereal day represents the time the earth takes to make one complete rotation about its axis. A sidereal day is equal to 23 hours 56 minutes 4.09 seconds of mean solar time.

The use of the rotation of the earth as a measurer of time is not without objection, for there can be no doubt that the mean solar day is gradually growing longer, due to the slowing down of the rotation of the earth. In order to remove this objection, it has been proposed to use the time of vibration of the atom of some element, such as sodium, as the > unit of time, for under definite conditions it appears as if this time were quite fixed and unalterable.

14. Units of Angular Measurement.-The ordinary unit adopted for measuring angles is the degree 90 degrees being equal to a right angle, so that 360 degrees correspond to a complete rotation. Each degree is divided into 60 minutes, and each minute into 60 seconds. Degrees, minutes, and seconds of arc are indicated by the symbols °, ', and" respectively.

Another unit of angle, which is frequently employed, is called the radian, and is such that if an arc of the circumference of a circle is taken equal in length to the radius of the circle, then this arc will subtend an angle at the centre which is equal to one radian. When the radian is used as the unit, the angle is said to be measured in circular measure.

If we have an arc of a circle, of which the length is a, then the angle subtended at the centre of the circle by this arc is equal to ar radians, where r is the radius of the circle. When a=r this of course reduces to one radian according to the definition.

Since the length of the circumference of a circle of radius r is 2πr, this arc will subtend 2′′r/r or 2 radians at the centre. But the angle subtended at the centre by the whole circumference is 360°.

Hence

Also

... I radian

= 360°/2π.

= 57°.2958.

= 57° 17′ 44′′.88.

I degree = 0.017453 radians.

If AB (Fig. 2) is an arc of a circle of radius r, described abo

C

FIG. 2.

B

point o as centre, then the an subtended by this arc at the

is equal to AB'r radians. If from A draw BC perpendicular to OA, th following trigonometrical relation good: 1

(in circular measure) = AB¦r.

Sin 0=BC|OB=BC|r.

Cos 0=OC/OB=OC/r.

Tan 0=BC|OC.

Now, if the angle is very small, the length of the arc AB very nearly the same as that of the perpendicular BC, while OC very nearly equal to OB or r. Hence, when is very small, the relations reduce to

0=AB r.

Sin 0=AB r.

Cos 0=r/r=1.

Tan 0=ABOA=AB'r.

Hence for small values of 0 we have

Sin 0=tan 0=0,

where is measured in radians, and

Cos 0=1.

The closeness with which these relations are true for different values of will be evident from the following table :

1 Readers unfamiliar with the elements of trigonometry may take these rela defining the quantities-the sine of the angle (written sin ), the cosine of and the tangent of @ (tan 0).

These relations will be found of considerable utility, for by their means we are able, whenever we are dealing with small angles, to considerably simplify many expressions involving these functions of the angle.

Since an angle is measured, in circular measure, by the ratio of the length of the arc (a) to the radius (r), we have, if [0] is taken to represent the dimensions of the unit angle, the relative

[0] = [Z] | [Z]

= 1.

Thus an angle has dimensions zero with reference to all the fundamental units. As the dimensions of any quantity cannot depend on the absolute value of the unit used to measure it, it follows that an angle, when measured in degrees, is also of zero dimensions with reference to the fundamental units of length, mass, and time.

A

CHAPTER III

MEASUREMENT OF LENGTH

15. Importance of Length Measurements.-In the measurement of nearly all physical quantities, what we actually observe is the ratio of some length to some other length. Thus when we measure the pressure of the atmosphere with a mercury barometer, what we really observe and measure is the length of a column of mercury; the same statement applies to the measurement of a temperature with a mercurial thermometer; so also, when we use a spring balance to measure a mass, it is the movement of a pointer along a scale that is observed. Hence we see the importance of being able to make accurate measurements of length.

With an ordinary scale divided into tenths of an inch it is possible, with a little care and practice, to measure by eye a length, which is not greater than that of the scale, to within one hundredth of an inch. This is done by mentally supposing each of the tenths of an inch subdivided into ten equal parts, i.e. into hundredths of an inch, and estimating by eye by how many of these imaginary hundredth of an inch divisions the length exceeds the nearest number of whole divisions. In the same way, with a scale divided into millimetres, it is possible to read to tenths of a millimetre. In order to attain to a degree of accuracy much greater than the above it is necessary to adopt some mechanical means of subdividing the divisions, for merely making the divisions of the scale nearer together does not advance matters much if we trust to our judgment and eye alone, even if a magnifying glass is used. Of such mechanical contrivances the most commonly employed are the vernier and the micrometer screw.

16. The Vernier.-Suppose AB, Fig. 3, is a scale divided into equal

10

E F G

FIG. 3.

1.5

20

parts, and that the end D of some object (CD), the length of which is to be measured, lies between the fourth and fifth divisions, and that we