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.

to B. Conceive the path of the ship to 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.; we shall thus have a series of triangles 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 (Geom., Prop. 18, B. 4,) Ab: Ab': bc bc':: cd: cd', &c.

therefore, (Geom., Prop. 6, B. 2,)

Β ́

B

ab : ab' : : ab + bc + cd + &c., : ab' + bc' + cd' + &c. But ab+bc + cd + &c., is the whole distance sailed, and ab' + bc' + cd' + &c. = AB', is the difference of latitude between A and B; consequently, if a right angled triangle a BB', similar to the small triangle abb', be constructed, that is, one in which the angle a is equal to the course, and if the hypothenuse a B represent 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 b'b + c'c + d'd + &c. of all the minute departures which the ship makes from the successive meridians which it crosses; for as the tri

A

angle ABB', in this last diagram, is similar to the small triangle abb', in the former, we have

to B.

Ab bb': bc: ec':: ed: dd', &c.

(1);

(2);

.. ab : bb' : : ab+bc'+ cd + &c. : bb'+cc'+ dd'+&c. ... 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 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 others may be found simply by the resolution of a right angled plane triangle; as 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.

EXAMPLES.

1. A ship from latitude 47° 30' N. has sailed S. W. by S. 98 miles. What latitude is she in, and what departure has she made?

Let c be the place sailed from, св the meridian, the angle c = 3 points = 33° 45', and CA 98 miles, the distance sailed; then CB will be the difference of latitude, and BA the departure. Hence by the formulæ for the solution of right angled triangles,

As rad.

As rad.

10.

1.991226

10.

: Distance 98 1.991226 : Dist.

:: cos.course 33°45' 9.919846:: sin. course

: Diff. of lat. 81.48 1.911072 : Departure 54.45 1.735965

Latitude left 47° 30' N.

Diff. of lat. 81.48 minutes 1 22 S. Dep. 54.45 miles W. Latitude in 46 8 N.

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 the S. and E., and the rate 5 miles an hour. Required the course, distance, and departure. Lat. left 38° 32′ N. Lat. in 36 56 N.

24 × 5 132 miles, the distance. 51

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, 938.9 miles W.

4. Two ports lie under the same meridian, one in latitude 52° 30′ N., and the other in latitude 47° 10′ N. A ship from the southernmost sails due east, at the rate of 9 miles an hour, and two days after meets a sloop which had sailed from the northernmost port; required the sloop's direct course and distance run.

Course S. 53° 28' E., or S. E. E.; distance run 537.6 miles. 5. If a ship from lat. 48° 27′ S. sail S. W. by W. 7 miles an hour, in what time will she arrive at the parallel of 50° S? In 23,914 hours. 6. If after a ship has sailed from lat. 40° 21′ N. to lat. 46°

18' N., she be found 216 miles to the eastward of the port left; required her course and distance sailed.

verse.

Course N. 31° 11' E., distance 417.3 miles.

Traverse Sailing.

98. When a ship, in going from one place to another, sails on different courses, it is called traverse sailing; and the determination of the single course and distance from the one place to the other is called working or compounding the traTo effect this, it is obviously merely necessary to find the difference of latitude, and departure, due to each distinct course, to take the aggregate of these for the whole difference of latitude and departure, and from these to find, as in last article, the single course and distance. It is usual in thus compounding courses to form a table consisting of six columns, called a traverse table, and in the first column to register the several component courses, and against them, in the second column, the proper distances; the next two columns, marked N. and S., ar to receive the several differences of latitude, whether N. or S., due to each course, and distance, and the two remaining columns marked E. and W. are to receive, in like manner, the corresponding eastings and westings, that is, the departures. When these several particulars are all inserted, the columns are added up, and the difference of the results of the N. and S. columns will be the required difference of latitude, and the difference of the results of the E. and W. columns will be the corresponding departure. (See page 134.)

The columns appropriated to the differences of latitude and departures are usually filled up from a table already computed to every quarter point of the compass, and to all distances from one mile up to 100 or 120; so that, by entering this table with any given course and distance, the proper difference of latitude and departure is found by inspection. Most books on navigation and also surveying, contain a second and more enlarged traverse table, being computed to every course from a quarter of a degree up to forty-five degrees. This latter