M

then

300. If RO be produced to cut the circle again in L,

=

TP2 = TR. TL

[Euc. III. 36]

· TR (RL + TR) = 2TR . RO + TR3.

But TR will in general be much less than a thousandth part of RO, and therefore TR will be much less than a thousandth part of 2TR. RO.

Hence, the formula TP2 = 2TR. RO, i.e. TP2 = twice the earth's radius × vertical height, will give the value of TP correct to at least three significant figures.

Example. Three times the height in feet of the place of observation above the sea is equal to twice the square of the distance of the horizon in miles.

Here, TP2=RL × 7914 miles.

Let f be the number of feet in RL, then the number of ; let x be the number of miles in TP, then 5280

miles in RL is

f

×7914= nearly. Q.E.D.

EXAMPLES. LXXV.

(1) Show that the limit of R2 sin

2π n

(i.e. the area of a poly

n
2

gon of n sides inscribed in a circle of radius R), when n= ∞ is

(3) Given that π=3·141592653589793...prove that the circular measure of 10" is '00004848136811...

(4) Prove that 2 sin (72o+A) – 2 sin (72o — A)=(√5 – 1) sin A, and that 2 sin (36°+4) − 2 sin (36o — A)=(√5+1) sin A. (5) If a mast of a ship be 150 feet high, show that the greatest distance seen from its top is 15 miles nearly.

(6) Prove that if the dip of the horizon at the top of a mountain is 1o 26′ [=tan-1·025], the mountain is about 6530 feet high.

NOTE. The definitions given in Arts. 75, 78 of the Trigonometrical Ratios are now used exclusively.

The NAMES tangent, secant, sine, were given originally to quantities defined as follows.

Let ROP be any angle. With centre O and any radius describe the arc RP. Draw PM perpendicular to OR and PT perpendicular to OP. (See Figure on previous page.)

Then PR is called an arc, PT is the tangent of the arc PR, OT is the secant of the arc PR, MP is the sine of the arc PR.

The name sine is derived from the word sinus. For, in the figure, PMP' is the string of the "bow" (arcus), and the string of a bow when in use is pulled to the archer's breast.

The co-tangent, co-secant and co-sine are respectively the tangent, secant and sine of the complement of the arc or of the angle.

The sine, tangent, etc. of the angle are the same as the measures of the sine, tangent, etc. of the arc, when the radius of the circle is the unit of length.

APPENDIX.

THE VERNIER, THE LEVEL, THE THEODOLITE, THE SEXTANT, THE MARINER'S COMPASS.

301. THE practical Surveyor† has to measure distances and angles, and has also to make plans or pictures, recording the result of his measurements.

For the measurement of distances the surveyor uses either rods, or chains, or tapes.

Rods used in measurement are made of wood, or of metal or sometimes (when extreme accuracy is required, as in the case of the measurement of the base line of the ordnance survey on Salisbury Plain) of glass.

All these instruments, when exposed to changes of temperature, are liable to change of length; hence for great accuracy, a surveyor must know the exact length of his measuring rod at all ordinary temperatures; and when making a measurement, must note the temperature of his rod at the instant of observation. The change of length caused by change of temperature is greater in a rod of metal than in a rod of wood. Hence wood is a very suitable material for measuring rods under ordinary circumstances. A tape made of cotton or hemp if used for measurement must be carefully protected from moisture by varnish or otherwise; as such tapes sensibly shrink when allowed to become damp; also, if of any considerable length, they stretch sensibly under tension.

A tape of 66 feet can be easily stretched an inch or so.

It must not be supposed that any verbal or pictorial description, such as the following, can in any way take the place of a practical explanation of the instruments themselves. A study of these figures may perhaps tell the student what to look for when he actually has the instrument in his hands.

THE VERNIER.

302. A vernier is a simple instrument for increasing the accuracy of the measurement of a small distance by one significant figure.

303. Description of a Vernier. Suppose we have a rule (i.e. a measuring rod) of brass graduated † to tenths of an inch.

The vernier is a little slip of brass which slides along the rule.

This slip of brass is a little more than 1

inches long, and a portion of its length 1 inches in length is divided into ten equal parts, by fine scratches on the surface of the metal.

11

Thus the distance between each scratch and the next is 10 of an inch or (+16) of an inch; i. e. this distance exceeds the distance between two scratches on the rule, by an hundredth part of an inch.

304. To read the Vernier. This will be best explained by an example.

Suppose the length to be measured is ascertained to be 3 ft. 11.5 inches and a little over.

This can be ascertained by the use of the rule (or measuring rod). Now let the rule be so placed that one end exactly coincides with one extremity of the length to be measured; then the other extremity K of the length to be measured will be between the scratches on the rule indicating 3 ft. 11.5 in. and 3 ft. 11.6 in. Now slide the CD vernier on the rule till its extremity D coincides with the extremity K of the length to be measured.

+ i.e. having fine scratches upon it, each the tenth part of an inch from the next one.

It will be observed that one of the scratches on the vernier coincides with one of the scratches on the rule more nearly than any other.

Suppose this to be the scratch marked 6 on the vernier. Then the length to be measured is

3 ft. 11.56 inches nearly.

For the length exceeds 3 ft. 11.5 inches by just as much as 6 spaces on the vernier exceed 6 tenths of an inch, that is by 6 hundredths of an inch.

305. A vernier may be used to read the graduations of a circular arc; in which case it is made curved so as to follow the line of the arc.

306. The student should notice that the advantage gained by the use of the vernier depends on the fact that the eye is able to judge with considerable accuracy when two scratches are or are not, coincident.

NOTE. French instrument makers make their verniers (10) of an inch and divide it into ten equal parts. We leave it as an exercise to the student to discover how such a vernier would be used.