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95. Having in the preceding parts of the present treatise pretty fully explained and illustrated the principles of plane and spherical trigonometry, we shall now, for the purpose

of showing the practical utility of these principles, apply them to the solution of one of the most important mathematical problems that ever has engaged the attention of man, viz. to determine the place of a ship at sea.

When a ship sails from any known place, and a correct account is kept of her various directions, and rates of sailing, her situation at any time may be readily ascertained by the rules of plane trigonometry, and the solution of the problem from these data belongs to Navigation.

But it is impossible to measure a ship's course and the distance sailed exactly ; so that after a long passage it would be unsafe to compute the place of the ship from the ship's reckoning In such cases, therefore, the solution must be effected from other data, independent of the ship's account; these are furnished by astronomical observation, and the computation is performed by the rules of spherical trigonometry; the problem then becomes one of Nautical Astronomy. We shall devote a distinet chapter to each of these important branches.





96. 1. The earth is very nearly spherical. For the purpos-a of Navigation it may be considered as perfectly so. It revolves round one of its diameters, called its axis, in about twentyfour hours. This rotation is from the west towards the east, causing the heavenly bodies to have an apparent motion from the east towards the west.

2. The great circle, whose poles are the extremities of the axis, is called the equator. The poles of the equator are called also the poles of the earth; the one being the north pole, and the other the south pole.

3. Every great circle which passes through the poles, and which, therefore, cuts the equator at right-angles, is called a meridian circle. Through every place on the surface of the earth such a great circle is supposed to be drawn; it is the meridian of the place. It is expedient for the purposes of Geography and Navigation to fix upon one of these meridians as a first meridian, from which the meridians of other places are measured.

The English have fixed upon the meridian of Greenwich Observatory for the first meridian.

4. The longitude of any place is the arc of the equator, intercepted between the meridian of that place and the first meridian; the longitude, therefore, is the measure of the

; angle between the two meridians. The longitude is east or west, according as the place is situated on the right or on the left of the first meridian, when we look towards the north pole.

5. The difference of longitude between two places is the arc of the equator, intercepted between the meridians of


those places, or the measure of the angle which they include; hence, when the longitudes of the places are of the same denomination, that is, either both east or both west, the difference is found by subtracting the one from the other; but when they are of contrary denominations the difference is found by adding the one to the other.

6. The latitude of a place is its distance from the equator, measured on the meridian of the place. Latitude, therefore, is north or south, according to the pole towards which it is measured, and cannot exceed 90°.

7. The small cireles drawn parallel to the equator, are called parallels of latitude. The arcs of a meridian, intercepted between two such parallels, drawn through any two places, measures the difference of latitude of those places : when the latitudes are of the same denomination, the difference of latitude is found by subtraction, but when the denominations are not the same, the difference of latitude is found by addition, like difference of longitude.

8. The horizon of any place is an imaginary plane, conceived to touch the surface of the earth at that place, and to be extended to the heavens; such a plane is called the sensible horizon, and one parallel to it, but passing through the earth's centre, is the rational horizon of the place, A line drawn across the horizon and through the place, in the plane of its meridian, is called a north and south line; the horizontal line through the same point, and perpendicular to this, is the east and west line. Besides the North, South, East and West, points thus marked on the boundary of the horizon, this boundary is conceived to be subdivided into other intermediate points, corresponding to the divisions in the circle on the next page.

9. The course of a ship is the angle which her track makes with the meridians; so long as this angle remained the same, if the meridians were all parallel, the path of the ship would be a straight line; but as the successive ones bend towards that from which the ship sets out as you approach the poles, the direction of her path is continually changing, and she


moves in a curve, called the rhumb line, or loxodromic curve. The magnitude of the angle or the course is indicated by the mariner's compass.

10. The mariner's compass consists of a circular card, whose circumference is divided into thirty-two equal parts, called points, and each of these is subdivided into four equal parts, called quarter points; across this card, and fastened to it, so that they move together, is fixed a slender bar of magnetized steel, called the needle; the tapering extremities of which point to two diametrically opposite divisions of the card. These opposite divisions are marked N. and S., corresponding to the north and south poles, or ends, of the mag netized bar. The diameter W.E., at right angles to the diameter N. S., point out the west and east points; these four are called the cardinal points, and the others are marked as in the subjoin


ed diagram.

Thus one point

from the north



towards the east, is north by east; two points, north, northeast; three points, north-east by north; and so on. quadrant contains eight points, so that a point is 90°+8= 11° 15'. (See Table of Rhumbs, Table V. at the end.)

The card thus furnished being now suspended horizontally, so as to move freely and allow the needle attached to it, to settle itself, will point out the four cardinal points of the horizon, as also the several intermediate points, provided only that it is the property of the magnetic needle to point due north and south. Such, however, is not strictly the case, as the needle is found, from accurate observations, to deviate

from this position, and at some places very considerably, and this deviation is itself subject to variation. But the true direction of the compass, or the angle it makes at any place with a line pointing duly north and south, may be ascertained at any time by astronomical observations, and thus the deviation of the compass points, from the corresponding points of the horizon, may always be found and allowed for.

The compass is so placed on ship-board that the vertical plane, cutting the ship from stem to stern, may pass through the centre of the card, so that that point of the compass which is directed to the ship's head shows the compass-course, and the proper correction for variation being applied, the true course will be obtained.

11. A ship's rate of sailing is determined by means of an instrument, called the log, and an attached line, called the log-line.

The log is a piece of wood, forming the sector of a circle, and its rim is so loaded with lead, that when heaved into the sea,

it assumes a vertical position, with its centre barely above the water. The logline is so attached as to keep the face of the log towards the ship, that it may offer the greater resistance to being dragged after the ship by the log-line, as it unwinds from a reel on board, by the advancing motion of the ship. The length of line thus unwound in half a minute, gives the rate of sailing. For convenience, the log-line is divided into equal parts, called knots, of which each measures the 120th of a nautical or geographical mile*, and as half a minute is the 120th of an hour, it follows that the number of knots, and parts of a knot, run in half a minute expresses the number of miles, and parts of a mile, run in an hour, at the same rate of sailing.


The geographical mile is one minute of the earth's circumference. Taking the diameter at 7916 English miles, the geographical mile will be about 6079 feet.

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