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 planes of the two meridians. The longitude is east or west, according as the place is situated east or west of the first meridian.

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 their planes 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 north or south of the equator, measured on the meridian of the place. Latitude cannot exceed 90°.

7. Small circles parallel to the equator, are called parallels of latitude. The arc of a meridian, intercepted between two such parallels, drawn through any two places, is the difference of latitude of those places; when the latitudes are of the same name, i. log both N. or both s., the difference of latitude is found by subtraction, but when not, the difference of latitude is found by addition.

8. The horizon of any place is an imaginary plane, touching the surface of the earth at that place, and extending to the heavens; such a plane is called the sensible horizon, and one parallel to it, but passing through the centre of the earth, the rational horizon of the place. The line of intersection of the plane of the horizon, and the plane of the meridian of a place, is called a north and south line; the horizontal line through the same point, and perpendicular to this, is called the east and west line. Besides the North, South, East, and West points, called cardinal points, thus determined on the boundary of the horizon, there are numerous subdivisions 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; if this angle continued the same, and the meridians were all parallel, the path of the ship would be a straight line; but as the meridians bend towards the poles, the direction of her path is continually changing, and she moves in a curve, called the rhumb line or loxodromic The instrument employed on ship-board to show the course of the ship is called the mariner's compass.

curve.

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, in the direction of a diameter, and fastened to the card, so that they move together, is fixed a slender bar of magnetized steel, called the needle; the 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 magnetized bar. The diameter E.W., at right angles to the diameter

N.S., points out the east and west points.

One point from the north towards the east, is marked N.E., and called north by east; two points, N.N.E., and called north north-east; three points,

north-east by north; and

so on. Each quadrant

contains eight points, so that a point is 90° 8 11° 15'. (See Table

of Rhumbs, Table XXVIII., 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 is found by comparison with astronomical observations. The card rests at its centre, on a pivot placed in the vertical plane, cutting the ship from stem to stern, and is held stationary in space by the magnetic forces of the earth, whilst the ship turns under it in changing her course, so that that point of the compass which is directed to the ship's head shows the ship's course, which must be corrected for the slight variation of the compass from the meridian, a variation which is different in different parts of the earth; the method of determining it will be hereafter given.

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 in the form of the sector of a circle, the rim of which is so loaded with lead, that when heaved into the sea it position, having its centre barely above the water.

assumes a vertical The log-line is so

206

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 logline, as it unwinds from a reel on board, by the advancing motion of the ship. The log-line is divided into equal parts, called knots, of which each measures the 120th of a nautical or geographical mile.* A half minute sand glass is used in connexion with the log. When the log is heaved, the instant the first knot on the line passes the hand of a sailor, the half minute glass is turned by a word, and the instant the sand is run out, the line is caught by a word; 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.

ON PLANE SAILING.

97. Let the annexed diagram represent a portion of the earth's surface, P the pole, and EQ the equator. Let A B be a rhumb line, or path described by a ship in sailing on a single course from A to B Let the rhumb line be divided into portions Ab, bc, cd, &c., so small that

each may differ insensibly

from a straight line, and draw meridians through these several divisions, as also the parallels of latitude bb', cc', dd', &c.; a series of triangles will thus be described on the surface of the globe, but so small that each may be considered as a plane triangle. These triangles are all similar, for the angles at b', c', d', &c., are right-angles, and the ship's path cuts all the meridians at equal angles; hence (Theorem 63 Geom.),

*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.

therefore, since by the theory of proportion the sum of the antecedents is to the sum of the consequents as any one antecedent is to its consequent,

ab: ab' :: ab + bc + cd + &c., : ab' + bc' + cd' + &c.

B'

B

But ab+be+cd+ &c., is the whole distance sailed, and ab'+ bc' + cả + đc. = AB', is the difference of latitude between A and B; consequently, if a right angled triangle ABB', similar to the small triangle abb' be constructed, that is, one in which the angle A is equal to the course, and the hypothenuse A B is equal to the distance sailed, the side A B' will represent the difference of latitude. Moreover, the other side BB', or that opposite to the course, will represent the sum bb + c'c + d'd + &c. of all the minute departures which the ship makes from the successive meridians which it crosses; for as the triangle ABB, in this last diagram, is similar to the small triangle abb', in the former we have

Ab: bb':: AB: BB'

but in the first figure we have

ab: bb' :: bc: cc' :: cd: dd', &c.

A

(1)

.. ab : bb' : : ab + bc + cd + &c. : bb'+ cc' + dd' + &c. (2)

consequently, since the three first terms of (1) are respectively equal to those of (2), the fourth term BB', of (1), must be equal to the fourth term, bb'+ cc'+ dd + of (2), &c. This last quantity is called the departure of the ship in sailing from A to B.* It follows, therefore, that the distance sailed, the difference of latitude made, and the departure, are correctly represented by the hypothenuse and sides of a right angled plane triangle, in which the angle opposite the departure is the course, so that when any two of these four things are given, the other two may be found simply by the resolution of a right angled plane triangle; so far, therefore, as these particulars are concerned, the results are the same as if the ship were sailing on a plane surface, the meridians being parallel straight lines, and the parallels of latitude cutting them at right angles; and hence that part of Navigation in which only distance sailed, departure, difference of latitude, and course are considered, is called Plane Sailing.

The departure is not to be confounded with BB' in the first diagram. It is greater than this, because the small departures bb', cc', &c., whose sum is the whole departure, lap over each other.

The two of the four elements which enter into problems of plane sailing usually given are course and distance, being found from observation.

EXAMPLES.

1. A ship from latitude 47° 30' N. has sailed S.W. by S. 98 miles. At what latitude has she arrived, and what departure has she made? Let c be the place sailed from, CB the meridian,

=

the angle c 3 points 33° 45', see Table of Rhumbs, and CA = 98 miles, the distance sailed; then CB will be the difference of latitude, and BA the departure. Then by the formulas for the solution of right angled triangles (forms (4) and (5) Art.

2. A ship sails for 24 hours on a direct course, from lat. 38° 32′ N, till she arrives at lat. 36° 56′ N.; the course is between S. and E., and the rate 5 miles an hour. Required the course, distance and departure. 24 × 5 = 132 miles, the distance.

Lat. left 38° 32′ N.

Hence the course is S. 43° 20' E., and the departure 90′58 miles E. 3. A ship sails from lat 3° 52′ S. to lat. 4° 30' N., the course being N.W. by W. W.; required the distance and departure.

Distance, 1065 miles; Departure 939.2 miles W. See last note but one. Miles are converted into degrees, &c., by dividing by 60.