CHAPTER IX

PLANE SAILING

125. Plane Sailing. A simple and interesting application of plane trigonometry is found in that branch of navigation in which the surface of the earth is considered a plane. This can be the case only when the distance is so small that the curvature of the earth may be neglected.

This chapter may be omitted if further applications of a practical nature are not needed.

126. Latitude and Departure. The difference of latitude between two places is the arc of a meridian between the parallels of latitude which pass through those places.

Thus the latitude of Cape Cod is 42° 2' 21" N. and the latitude of Cape Hatteras is 35° 15' 14" N. The difference of latitude is 6° 47′ 7′′.

The departure between two meridians is the length of the are of a parallel of latitude cut off by those meridians, measured in geographic miles.

1

360

The geographic mile, or knot, is the length of 1′ of the equator. Taking the equator to be 131,385,456 ft., of of this length is 6082.66 ft., and this is generally taken as the standard in the United States. The British Admiralty knot is a little shorter, being 6080 ft. The term "mile" in this chapter refers to the geographic mile, and there are 60 mi. in one degree of a great circle.

Calling the course the angle between the track of the ship and the meridian line, as in the case of N. 20° E., it will be evident by drawing a figure that the difference in latitude, expressed in distance, equals the distance sailed multiplied by the cosine of the course. That is diff. of latitude distance x cos C'.

=

In the same way we can find the departure. This is evidently given by the equation

For example, if a ship has sailed N. 30° E. 10 mi., the difference in latitude, expressed in miles, is

and the departure is 10 sin 30° 10 x 0.5 5.

:

=

=

127. The Compass. The mariner divides the circle into 32 equal parts called points. There are therefore 8 points in a right angle, and a point is 11° 15'. To sail two points east of north means, therefore, to sail 22° 30' east of north, or northnortheast (N.N.E.) as shown on the compass. Northeast (N.E.) is 45° east of north. One point east of north is called north by east (N. by E.) and one point east of south is called south by east (S. by E.). The other terms used, and their significance in angular measure,

will best be understood from the illustration and the following table:

The compass varies in different parts of the earth; hence, in sailing, the compass course is not the same as the true course. The true course is the compass course, with allowances for variation of the needle in different parts of the - earth, for deviation caused by the iron in the ship, and for leeway, the angle which the ship makes with her track.

Exercise 64. Plane Sailing

1. A ship sails from latitude 40° N. on a course N.E. 26 mi. Find the difference of latitude and the departure.

2. A ship sails from latitude 35° N. on a course S.W. 53 mi. Find the difference of latitude and the departure.

3. A ship sails from a point on the equator on a course N.E. by N. 62 mi. Find the difference of latitude and the departure.

4. A ship sails from latitude 43° 45' S. on a course N. by E. 38 mi. Find the difference of latitude and the departure.

5. A ship sails from latitude 1° 45' N. on a course S.E. by E. 25 mi. Find the difference of latitude and the departure.

6. A ship sails from latitude 13° 17'S. on a course N.E. by E. & E., until the departure is 42 mi. Find the difference of latitude and the latitude reached.

7. A ship sails from latitude 40° 20' N. on a N.N.E. course for 92 mi. Find the departure.

8. If a steamer sails S.W. by W. 20 mi. what is the departure and the difference of latitude?

9. If a sailboat sails N. 25° W. until the departure is 25 mi., what distance does it sail?

10. A ship sails from latitude 37° 40′ N, on a N.E. by E. course for 122 mi. Find the departure.

11. A yacht sails 6 points west of north, the distance being 12 mi. What is the departure?

12. A steamer sails S.W. by W. 28 mi. It then sails N.W. 30 mi. How far is it then to the west of its starting point?

13. A ship sails on a course between S. and E. 24 mi., leaving latitude 2° 52' S. and reaching latitude 2° 58' S. Find the course and the departure.

14. A ship sails from latitude 32° 18′ N., on a course between N. and W., a distance of 34 mi. and a departure of 10 mi. Find the course and the latitude reached.

15. A ship sails on a course between S. and E., making a difference of latitude 13 mi. and a departure of 20 mi. Find the distance and the course.

16. A ship sails on a course between N. and W., making a difference of latitude 17 mi. and a departure of 22 mi. Find the distance and the course.

128. Parallel Sailing. Sailing due east or due west, remaining on the same parallel of latitude, is called parallel sailing.

129. Finding Difference in Longitude. In parallel sailing the distance sailed is, by definition (§ 126), the departure. From the departure the difference in longitude is found as follows: Let AB be the departure. Then in rt. A OAD

The triangles DAB and OEQ are similar, the arcs being (§ 125) considered straight lines.

AB

EQ

That is, the number of minutes in the difference in longitude is the product of the number of miles in the departure by the secant of the latitude, the nautical, or geographic, mile being a minute of longitude on the equator.

Exercise 65. Parallel Sailing

1. A ship in latitude 42° 16' N., longitude 72° 16' W., sails due east a distance of 149 mi. What is the position of the point reached?

2. A ship in latitude 44° 49' S., longitude 119° 42' E., sails due west until it reaches longitude 117° 16' E. Find the distance made.

3. A ship in latitude 60° 15' N., longitude 60° 15' W., sails due west a distance of 60 mi. What is the position of the point reached?

130. Middle Latitude Sailing. Since a ship rarely sails for any length of time due east or due west, the difference in longitude cannot ordinarily be found as in parallel sailing (§§ 128, 129). Therefore, in plane sailing the departure between two places is measured generally on that parallel of latitude which lies midway between the

parallels of the two places. This is called the method of middle latitude sailing. Hence, in middle latitude sailing,

This assumption produces no great error, except in very high latitudes or excessive runs.

Exercise 66. Middle Latitude Sailing

1. A ship leaves latitude 31° 14' N., longitude 42° 19' W., and sails E.N.E. 32 mi. Find the position reached.

2. Leaving latitude 49° 57' N., longitude 15° 16' W., a ship sails between S. and W. till the departure is 38 mi. and the latitude is 49° 38' N. Find the course, distance, and longitude reached.

3. Leaving latitude 42° 30' N., longitude 58° 51' W., a ship sails S.E. by S. 48 mi. Find the position reached.

4. Leaving latitude 49° 57' N., longitude 30° W., a ship sails S. 39° W. and reaches latitude 49° 44' N. Find the distance and the longitude reached.

5. Leaving latitude 37° N., longitude 32° 16' W., a ship sails between N. and W. 45 mi. and 'reaches latitude 37° 10' N. Find the course and the longitude reached.

6. A ship sails from latitude 40° 28' N., longitude 74° W., on an E.S.E. course, 62 mi. Find the latitude and longitude reached.

7. A ship sails from latitude 42° 20' N., longitude 71° 4' W., on a N.N.E. course, 30 mi. Find the latitude and longitude reached.