student must be cautioned that in most of the experiments on magnet poles similar perturbing causes are at work. The magnetism in a magnet is not quite fixed, but is liable to be disturbed in its distribution by the near presence of other magnet poles, for no steel is so hard as not to be temporarily affected by magnetic induction.

NOTE ON WAYS OF RECKONING ANGLES AND SOLID ANGLES

180

90°

144. Reckoning in Degrees. — When two straight lines cross one another they form an angle between them; and this angle may be defined as the amount of rotation which one of the lines has performed round a fixed point in the other line. Thus we may suppose the line CP in Fig. 88 to have originally lain along CO, and then turned round to its present position. The amount by which it has been rotated is clearly a certain fraction of the whole way round; and the amount of rotation round C we call "the angle which PC makes with OC," or more simply "the angle PCO." But there are a number of different The ways of reckoning this angle. common way is to reckon the angle by "degrees" of arc. Thus, suppose a circle to be drawn round C, if the circumference of the circle were divided into 360 parts each part would be called "one degree" (1°), and the angle would be reckoned by naming the number of such degrees along the curved arc OP. In the figure the arc is about 571°, or

270 Fig. 88.

571 360

of the whole way round, no matter what size the circle is drawn. 145. Reckoning in Radians. A more sensible but less usual way to express an angle is to reckon it by the ratio between the length of the curved arc that "subtends" the angle and the length of the radius of the circle. Suppose we have drawn round the centre C a circle whose radius is one centimetre, the diameter will be two centimetres. The length of the circumference all round is known to be about 3 times the length of the diameter, or more exactly 3.14159. This number is so awkward

...

that, for convenience, we always use for it the Greek letter ". Hence the length of the circumference of our circle, whose radius is one centimetre, will be 6-28318 ... centimetres, or 2 centimetres. We can then reckon any angle by naming the length of arc that subtends it on a circle one centimetre in radius. If we choose the angle PCO, such that the curved arc OP shall be just one centimetre long, this will be the angle one, or unit of angular measure, or, as it is sometimes called, the angle PCO will be one "radian." In degree-measure one radian 57° 17' nearly.

=

360°

2π

=

All the way round the circle will be 2 radians. A right angle will be radians.

π

2

P/

A

C

Fig. 89.

M

0

- In trigonometry

146. Reckoning by Sines or Cosines. other ways of reckoning angles are used, in which, however, the angles themselves are not reckoned, but certain "functions" of them called "sines," 66 cosines," tangents," etc. For readers not accustomed to these we will briefly explain the geometrical nature of these "functions." Suppose we draw (Fig. 89) our circle as before round centre C, and then drop down a plumb-line PM, on to the line CO; we will, instead of reckoning the angle by the curved arc, reckon it by the length of the line PM. It is clear that if the angle is small, PM will be short; but as the angle opens out towards a right angle, PM will get longer and longer (Fig. 90). The ratio between the length of this line and the radius of the circle is called the "sine" of the angle, and if the radius is 1 the length of PM will be the value of the sine. It can never be greater than 1, though it may have all values between 1 and -1. The length of the line CM will also depend upon the amount of the angle. If the angle is small CM will be nearly as long as CO; if the angle open out to nearly a right angle CM will be very short. The length of CM (when the radius is 1) is called the "cosine" of the angle. If the angle be called e, then we may for shortness write these functions:

90 P

M

Fig. 90.

60

30

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:

M

Fig. 91.

148. Solid Angles.- When three or more surfaces intersect at a point they form a solid angle: there is a solid angle,

P

N

E

F

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 subtended at P by the surface EF. To reckon this solid angle we adopt an expedient similar to that adopted when we wished to reckon a plane angle in radians. About the point P, with radius of 1 centimetre, describe a sphere, which will intercept

Fig. 92.

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 2. 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.

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 thereLore 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.

Gilbert made the great

150. The Earth a Magnet. 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 a horizontally-suspended compass-needle.