An angle is measured by an arc of the circle, of which the point is the centre; thus the angle F C A (Fig. 1) is measured by the arc F ▲; it is equally measured by the inner dotted arc, since both arcs contain the same number of degrees. To express this in another form, we say that any angle contains the same number of degrees, minutes, &c., as are contained in the arc which subtends it; and from this it also follows that increasing or diminishing the length of the legs does not in any way change the measure of the angle; if we extended the legs oF and o A to the length of a foot or more, and from the centre o described between the legs an arc of a circle with a radius of a foot, that arc would still be the measure of the angle F O A; neither arc nor angle would have increased, for the arc would still bear the same proportion to the circle of which it formed a part, as the arc F A does to the circle A B D E A.

When the arc that subtends an angle is a quadrant (90°), as the arc ▲ B, then the angle, as в O A, is a right angle, also of 90°, and the two legs are perpendiculars; an acute angle contains less than 90°, and an obtuse angle more than 90°.

To measure or to lay off angles and distances we use

THE PROTRACTOR AND DIAGONAL SCALE.

The protractor is a small semicircle of brass or horn, the circumference of which is neatly divided into degrees, and the centre of the straight edge is marked by a dot or small notch. It serves not only to draw angles, but to measure those already drawn. To measure an angle, let the notch or dot be placed on the angular point, and make the edge coincide with one of the lines that contains the angle; then the number of degrees cut off by the other line, computing on the semicircular part of the protractor, will show the value of the angle that has been measured.

The horn (and consequently transparent) protractor represented on Fig. 3 of Plate I. is marked to degrees, and points of the compass. It is very useful in chart work if a hole be drilled where the dot stands, and a length of thread be inserted therein; then, courses can be marked off, and bearings laid down, without the aid of the ordinary parallel rule. An angle is measured by it in a slightly different way from that described above; as an example, we will measure the angle F C A on Fig. 1, p. 9: place the dot over the point at c, and also the line extending from this dot to o (on the circumference of the protractor) over the line C A; then it will be seen that the line c F in the fig. coincides with the line on the protractor extending to 45°; hence the angle F C A measures 45°. In a similar manner, with the edge of the protractor over D A, and the dot over c, the line from the dot to o is over c B, and the protractor shows that the angle B C G is 7 points or 7820, and the angle G C D measures 1 point.

The oblong ivory protractor is marked in degrees along three sides, but it is used in the same manner as the semicircles. There are other scales on it, as the scale of chords (marked Cho. in Fig. 4, Plate I.), which is also used to measure and lay off angles; and there is the diagonal scale of equal parts (Fig. 2) to measure distance.

I will explain the use of the diagonal scale, taking the top row of figures. the 1, 2, 3, &c., may be taken as units, tens, or hundreds; if as units, then the divisions from B to A will represent decimals; if as tens, the same divisions will be units; and if as hundreds, then these divisions will be tens. Suppose, in drawing a geometrical figure, you have to lay off 58 miles; in this case put

one point of the dividers in 5 (for 50) and extend the other point to the 8th division from в in the direction of A, and you have 58; the bottom row of figures is for the same purpose, but the smaller sub-divisions are taken in the direction F; and remember that in the same projection you cannot change the value of the scale, now calling it hundreds, and now tens. You will better understand the use of this scale when we project the traverse of a day's run.

THE SHIP'S PLACE AT SEA.

LATITUDE AND LONGITUDE.

Our next advance is towards a knowledge of what is meant by a SHIP's PLACE AT SEA.

The earth as a sphere revolving on an imaginary axis has two poles, which

are the ends of the axis. That pole which is adjacent to Europe is called by common consent the NORTH POLE, and the other is the SOUTH POLE. Now, any circle that divides a sphere into two equal parts is a great circle, and of these there are many around our earth, as well as small circles and other lines-imaginary, no doubt, but mathematically as real as if the whole had an actual existence.

The principal great circle passing round our earth is the EQUATOR (E Q, Fig. 2), which is midway between the poles, and 90° from each of them; it divides the sphere into two equal parts-the northern hemisphere and the southern hemisphere-and is the starting point from which LATITUDE is reckoned, so that any place on the Equator is said to be in Lat. 0°; the N. pole is, therefore, in Lat. 90° N., and the S. pole in Lat. 90° S.; any intermediate place between the equator and a pole must have its latitude something between 0 and 90°, and it will be N. or S. according to whether it be in the Northern or Southern hemisphere. In Fig. 2 the lines parallel with the equator represent small circles called PARALLELS, and each indicates a given distance from the equator, and as in this case they are 15° apart, any place on the small circle a b would be said to be on the parallel of 60° N., or otherwise in Lat. 60° N.

In our figure, the circles which meet in the poles N and s are the great circles called MERIDIANS, and they are all perpendicular to the equator; by their aid LONGITUDE is reckoned. It has been shown that the starting point for latitude is the equator; there is no such well defined position for longitude, hence different nations chose different meridians as the initial point of reckoning; we select Greenwich; the French, Paris; the Spaniards, San Fernando; and so on. The first meridian being determined on (as Greenwich, indicated, let us say, by the thick line in Fig. 2), then, looking north, any place on a meridian to the right of that of Greenwich is said to be in East Long.; any place to the left of

* So called because at noon or midday the sun is on the meridian of a place.

the same meridian is in West Long.; and longitude is reckoned in this manner E. or W. through 180°, which is the greatest longitude a place can have. The meridians in the figure are drawn 15° apart, therefore a place situated anywhere, in either hemisphere, on the first circle to the left of the thick circle (Greenwich meridian) would be in Long. 15° W., as a place on the first circle to the right of Greenwich meridian, would be in Long. 15o E., and

80 on.

Thus we define Longitude as an arc of the equator (because it is measured along the equator) between the meridian of a fixed station and the meridian passing through any given place, reckoned E. or W. to 180°; and we define Latitude as an arc of the meridian between the equator and any given place, because it is measured along the meridian, reckoned in a N. or S. direction.

But if we know the Lat. of any place and not the Long., or, on the other hand, the Long, and not the Lat., its position on the earth is not determined, because, if we say it is in Lat. 42° N., it may be anywhere on the parallel of 42° N. in E. or W. Long., in the Pacific or in the Atlantic Ocean; similarly, if we say a place is in Long. 32° E., it may be in Europe, in Africa, in the Mediterranean, or in the seas south of the Cape of Good Hope. Hence both Lat. and Long are required to fix the position; and when we say that a ship is in Lat. 46° 20′ N., Long. 32° 40′ W., we have its PLACE AT SEA accurately noted—a spot in the Atlantic-and a course can be shaped therefrom towards the port of destination. It is part of the routine of a voyage to determine the ship's place day by day, approximately by dead reckoning, and more accurately by observation of the heavenly bodies.

We are now in a position to take the Mariner's Compass in hand and understand its indications.

THE MARINER'S COMPASS.

When out of sight of land, the MARINER'S COMPASS is the only instrument that shows the DIRECTION in which the ship is moving; it is therefore necessary to understand its construction and use, together with the corrections which must be applied to its indications.

In the first place you must learn the divisions of the compass card, and know how to box the compass, as the saying is. (See Compass, p. 14.)

The Compass Card is a circle divided into 32 parts, called points of the compass, and standing on the deck of a vessel at sea you must always suppose yourself to be in the centre of such a circle, the circumference of which is the visible horizon, and towards some point of which the vessel's head is directed. The standard of direction is the meridian passing through the place of the vessel, and a small part of this meridian is represented on the compass by a line drawn from one part of the circumference to another, but passing through the centre of the circle; one end of the meridional line is named N. (north), because the line trends in direction towards the north pole; and the other end, directed towards the south pole, is named S. (south). A second line, passing through the centre of the circle, but at right angles to the meridian, bas one end named E. (east), and the other W. (west); when facing the North, the East (in which direction the sun and other heavenly bodies rise) is on your right hand, and the