T

147. Reckoning by Tangents. Suppose we draw our circle as before (Fig. 91), but at the point O draw a straight line touching the circle, the tangent line at O; let us also prolong CP until it meets the tangent line at T. We may measure the angle between OC and OP in terms of the length of the tangent OT as compared with the length of the radius. Since our radius is 1, this ratio is numerically the length of OT, and we may therefore call the length of OT the "tangent" of the angle OCP. It is clear that smaller angles will have smaller tangents, but that larger angles may have very large tangents; in fact, the length of the tangent when PC was moved round to a right angle would be infinitely great. It can be shown that the ratio between the lengths of the sine and of the cosine of the angle is the same as the ratio between the length of the tangent and that of the radius; or the tangent of an angle is equal to its sine divided by its cosine. The formula for the tangent may be written:

C M

Fig. 91.

sect at a point they form a solid angle: there is a solid angle,

for example, at the top of a pyramid, or of a cone, and one at every corner of a diamond that has been cut. If a surface of any given shape be near a point, it is said to subtend a certain solid angle at that point, the solid angle being mapped out by drawing lines from all points of the edge of this surface to the point P (Fig. 92). An irregular cone will thus be generated whose solid angle is the solid angle subTo reckon this solid angle we adopted when we wished About the point P, with

that

tended at P by the surface EF. adopt an expedient similar to to reckon a plane angle in radians. radius of 1 centimetre, describe a sphere, which will intercept

the cone over an area MN: the area thus intercepted measures the solid angle. If the sphere have the radius 1, its total surface is 4. The solid angle subtended at the centre by a hemisphere would be 2π. It will be seen that the ratio between the area of the surface EF and the area of the surface MN is the ratio between the squares of the lines EP and MP. The solid angle subtended by a surface at a point (other things being equal) is inversely proportional to the square of its distance from the point. This is the basis of the law of inverse squares.

A table of radians, sines, tangents, etc., is given at the end of this book as Appendix A.

149. The Mariner's Compass. It was mentioned in Art. 87 that the compass sold by opticians consists of

a magnetized steel needle balanced on a fine point above a card marked out N, S, E, W, etc. The Mariner's Compass is, however, somewhat differently arranged.

In Fig. 93 one of the forms of a Mariner's Compass,

used for nautical observations, is shown. Here the card, divided out into the 32 points of the compass," is itself attached to the needle, and swings round with it so that the point marked N on the card always points to the north. In the best modern ships' compasses, such as those of Lord Kelvin, several magnetized needles are placed side by side, as it is found that the indications of such a compound needle are more reliable. The iron fittings of wooden vessels, and, in the case of iron vessels, the ships themselves, affect the compass, which has thereore to be corrected by placing compensating masses of iron near it, or by fixing it high upon a mast. The error of the compass due to magnetism of the ship is known as the deviation.

150. The Earth a Magnet. Gilbert made the great discovery that the compass-needle points north and south because the earth is itself also a great magnet. The magnetic poles of the earth are, however, not exactly at the geographical north and south poles. The magnetic north pole of the earth is more than 1000 miles away from the actual pole, being in lat. 70° 5' N., and long. 96° 46′ W. In 1831, it was found by Sir J. C. Ross to be situated in Boothia Felix, just within the Arctic Circle. The south magnetic pole of the earth has never been reached; and by reason of irregularities in the distribution of the magnetism there appear to be two south magnetic polar regions.

151. Declination. In consequence of this natural distribution the compass-needle does not at all points of the earth's surface point truly north and south. Thus, in 1894, the compass-needle at London pointed at an angle of about 17° west of the true north; in 1900 it will be 16° 16'. This angle between the magnetic meridian * and

* The Magnetic Meridian of any place is an imaginary plane drawn through the zenith, and passing through the magnetic north point and magnetic south point of the horizon, as observed at that place by the pointing of horizontally-suspended compass-needle,

the geographical meridian of a place is called the magnetic Declination of that place. The existence of this declination was discovered by Columbus in 1492, though it appears to have been previously known to the Chinese, and is said to have been noticed in Europe in the early part of the thirteenth century by Peter Peregrinus. The fact that the declination differs at different points of the earth's surface, is the undisputed discovery of Columbus.

In order that ships may steer by the compass, magnetic charts (Art. 154) must be prepared, and the declination at different places accurately measured. The upright pieces P, P', on the "azimuth compass" drawn in Fig. 93, are for the purpose of sighting a star whose position may be known from astronomical tables, and thus affording

CLAPL

Fig. 94.

a comparison between the magnetic meridian of the place and the geographical meridian, and of measuring the angle between them.

152. Inclination or Dip. Norman, an instrument - maker, discovered in 1576 that a balanced needle, when magnetized, tends to dip downwards toward the north. He therefore constructed a Dipping-Needle, capa

angle of 71° 50′.

A simple form of dipping-needle is shown in Fig. 94. The dip-circles used in the magnetic observatory at Kew are much more exact and delicate instruments. It was, however, found that the dip, like the declination, differs at different parts of the earth's surface, and that it also undergoes changes from year to year. The "dip" in London for the year 1894 is 67° 18′; in 1900 it will be 67° 9'. At the north magnetic pole the needle dips straight down. The following table gives particulars of the Declination, Inclination, and total magnetic force at a number of important places, the values being approximately true for the year 1900.

TABLE OF MAGNETIC DECLINATION AND INCLINATION (for Year 1900)

order to specify exactly the magnetism at any place;

these three elements are: