called degrees, written thus, 360°; each degree is divided into 60 equal parts, called minutes, written thus, 60'; each minute is divided into 60 equal parts, called seconds, written thus, 60". Any line drawn from the centre of a circle to its circumference is called a radius. Any line drawn across the circle from side to side through the centre is called a diameter, and is, of course, equal to twice the radius. Any two lines drawn from the same point contain an angle. Any two lines drawn from the centre of a circle to the circumference contain an angle, and the size of the angle is measured by the number of degrees between the two lines on the circumference. Thus in Fig. 73 the two lines A B and A D contain an angle, C, which is measured on the circumference, and is there seen to be about 26, or more exactly 26° 34'. If from the point D (in the line A D) a line, D E, is drawn vertically to and touching

and A is called and cosines can When an angle of a right angle That part of the

the line A B, the line D E is called the sine of the angle C. That part of the line A B which is between E the cosine of the angle C. In the same way, sines be drawn for any other angle not exceeding 90°. has 90° it is called a right angle; the two sides are perpendicular to, or square to, each other. line A B which is between E and B is called the versed sine of the angle C. That part of the circumference which is between B and D is called the arc. If from B a line is drawn perpendicular to A B until it meets the line A D produced to F, the line B F is called the tangent of the angle C, and the line A F the secant of the angle. If any number of degrees are taken from a quadrant, the number remaining to make up 90 are called the complement. This complement has also a sine, tangent, and secant, which are called the cosine, cotangent, and cosecant. That part of

the radius which is equal in length to the sine of the complement is commonly called the cosine, as shown above (see Fig. 73).

If the line A D is taken to represent the inclination of a seam of coal, and the line A B the level line, then the arc measures the angle of inclination, and it is found to be 26° 34'. Then the line D E, which is the sine, represents the vertical fall or rise of the slope A D; and the line A E, which is the cosine, represents the horizontal length between the points A and D. Therefore the vertical rise or fall horizontal length :: sine cosine; or horizontal length cosine

=

vertical (rise or fall) sine

The value of this rule (the truth of which is self-evident) is to be found in the fact that the relative lengths of the sines and cosines of every angle from o° to 90° have been carefully calculated, and are given in every book of mathematical tables.

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If the reader has such a book, and will look under the head of "Natural Sines, Cosines, etc.," he will find a page headed " 26°," and under this figure he will find, rather more than halfway down the column, 34'; opposite this latter figure, and in the column headed Sine," he will find the figures 4472388; under the column headed "Cosine" he will find the figures 8944146. These figures represent cosine 8944146 the relative lengths of the sine and cosine; or sine 4472388; 8944 2

=

=

= ; 4472 I

that

or, striking out the last three figures from each, is to say, the horizontal length is twice as much as the vertical fall (or rise); or the slope is 1 in 2, or 50 per cent.

Instead of using the table of sines and cosines, we may calculate in another way. Assume that the line A F, or secant, represents the inclined surface; then the line F B, or tangent, represents the vertical rise or fall, and the line A B represents the horizontal length between the points A and F. The line A B is the radius. Referring now to a book of mathematical tables, under the head of "Natural Tangents and Secants," we shall not find the proportional length of the radius given, because it is always taken as one; but under the head of "Tangents," and under the column " 26°," and opposite the figure 34 in column of minutes, will be found the radius figure 0'5000352; therefore tangent

I

0'5000352

; or, leaving out

0'5000 0'5

distance which is radius is twice the vertical distance which is

tangent; or the slope is 1 in 2, or 50 per cent. (see p. 49 for use of cotangent).

In case there are no mathematical tables at hand, the angle may be marked out by a protractor. If there is no protractor at hand, a circle may be drawn with a bow-pen (or pair of com

CHORD

AN EQUIANGULAR
AND

EQUILATERAL
TRIANGLE

CHORD

120°

10°

SLOPE

HORIZONTAL

passes), and divided into quadrants; each quadrant may be bisected, and then divided. in 45 equal parts, each of which will be a degree. Another and an easier way is to draw a circle or part of a circle, and with the compasses open at the same length as radius, mark off chords on the circumference (see Fig. 74). A chord equal in length to the radius covers an arc of 60°; so that six such chords complete the circumference. Each arc, therefore, covers 60°; if it is bisected, each division contains 30°; this can easily be divided into three nearly equal parts each of 10°, and these again into single degrees.

FIG. 74.-Circle divided by chords.

The slope being set out at the ascertained angle, 20°, as A B (Fig. 74), for the length of the incline, say 250 yards, represented by the length A B, then BC can be drawn vertical to AC the horizontal line, and BC and AC may then be measured with a scale, and are approximately 85 yards and 235 yards, or to find the percentage; or, as 235: 85 :: 100 x; x = 36; or the slope is 36 per cent. On looking at the table of tangents, it will be seen that the exact percentage is 36.39702.

8500

235

=

The equipment of the explorer should contain a good hammer, a magnifying-glass, a small bottle of acid to test for limestone, a clinometer, an aneroid barometer and thermometer to measure altitudes, a compass, a note-book, and a map. For detailed explorations, the services of stout labourers with shovels, pickaxes, drills, blasting powder (dynamite, etc.), boring-tools, etc., will be required.

53

CHAPTER III.

BORING: HAND, STEAM, RODS, ROPES, TUBES, FREE-FALL,

DIAMOND, ETC.

BORE-HOLES are in many cases the best means of completing the exploration of mineral estates.

A pit shows the ground more plainly, but in the majority of situations a pit cannot be sunk many feet deep before water finds its way in, and then the cost of sinking becomes too great for the purposes of exploration. Apart altogether from water, the cost of making a small hole from 3 inches up to 12 inches in diameter is generally very much less than the cost of sinking a large pit from 6 feet to 16 feet in diameter. The bore-hole may also be put down more quickly.

Two Kinds of Boring.-There are two chief modes of boring. One is by a percussive drill, which chips the rock into small fragments, subsequently removed; and the other, by a rapidly revolving ring, which grinds the rocks into powder. Of the percussive method of boring there are many varieties. Of the grinding method there are only two varieties; one of these is the diamond drill, and the other is a similar tool, but using hardened steel instead of diamonds.

Percussive Boring.—Various Methods.-The most common method of boring in England for depths of from 5 to 50 yards (a method used in some cases for depths of several hundred yards) is by means of a steel chisel screwed on to iron rods, suspended by a spring-pole. The chisel is steel welded on to the end of an iron rod, making a total length of 18 inches. It is a single blade, for hard rock a cross-blade, as wide as the intended diameter of the hole, say from 3 inches up to 8 inches. Unless it is intended to go very deep, the holes are seldom started more than 6 inches in diameter.

The rod varies from inch square up to 1 inch square; inch and 1 inch are common sizes. The rods must be made of the very best quality of iron that can be obtained, so as to diminish the chances of fracture. They are screwed together the

54

socket end downwards. There are two sets of shoulders at the upper end of each rod, on opposite sides (see Fig. 75, 1, 2, 3, 23).

FIG. 75.-Hand-boring tools. 1, single chisel; 2, cross chisel; 3, rod; 4, sludger (6 ft.); 5, screw-jointed tube; 6, tube with outside collar; 7, tube with flush joint riveted; 8, tube with screw socket; 8a, riveted socket joint; 9, grappler (6 ft.); 10, reamers; 11, single cross-head; 12, double cross-head; 13, fork; 14, key; 15, 16, core cutter and extractor; 17, spring-pole; 18, windlass; 19, legs and pulleys; 20, hook to lift rods; 21, temper screw; 22, excavation and hole; 23, rods; 24, core-cutter; 25, rods jointed with loose socket.

By means of these shoulders the rod can be lifted or suspended.

1 These flush-jointed tubes are the same diameter inside and outside through the whole length of tubing.