the angle on the bow has doubled or the object bears N E, again note the log. The distance a' e, or the distance of the ship from the object, is then equal to the distance a a' run in the interval, because

This method is most frequently used when the ship is at a', or when the object bears 4 points on the bow, and when the. ship arrives at b, or the object is exactly abeam, the distance be is equal to a' b. For this reason it is known among navigators as the four-point bearing. The method is a very useful one in coastwise navigation and should be frequently practiced.

16. Attention is called to the fact that when the ship arrives at a', Fig. 8, a' b and also be can be found by multiplying a' e ( = a a') by .71, or, as is done in practice, by .7. Thus, if a' e is equal to 5 miles, a' b and be will each be equal to .7 X 5 3.55, or 3.5, miles, the latter being preferred, as it places the ship nearer the danger than she actually is. From "General Navigation" (a British Admiralty publication), the following remarks are made in reference to this method: "In passing near a point of land or an island, the method of fixing a ship's position by doubling the angle on the bow is invaluable. The ordinary form of this method, the so-called four-point bearing, when the bearing is taken 4 points on the bow and on the beam, gives an excellent fix, but does not insure safety, as the point or the rocks off the object are abeam before the position is obtained. By taking the bearings 2 points and 4 points on the bow, however, as in Fig. 8, a very good position is obtained before the object is passed, the distance of the latter at the second bearing being, as shown, equal to the distance run in the interval."

The student should notice that this method is applicable, also, after the ship has passed the object. In such a case the

first observed angle on the stern should be twice the second angle. Thus, if the angle between the object and the ship's stern is 4 points, the second angle or bearing should be noted when 2 points. The distance run between the bearings or readings of the patent log is then equal to the distance of the ship from the object when it bore 4 points.

NOTE. - When practicing this method, it is well to bear in mind that as long as the course remains the same it is unnecessary to apply deviation or other corrections on the bearing, the compass in such a case serving simply as an instrument to measure the apparent angles. The exactness of the position found by this method will depend on the accuracy of recording the bearings and the distance run in the interval. For vessels engaged in coastwise navigation, a good idea in connection with this method is to have a large compass rose painted on the bridge or on deck in the place usually occupied by the officer in charge. The bearings of any object may then be very conveniently taken and observed by the officer standing in the center of the rose, thus eliminating the use of the compass.

17. By Means of Two Bearings of the Same Object. — It is evident that the foregoing method is subject to certain restrictions. You must wait and watch until the angle on the bow has doubled, or, if the ship has passed the object, until the angle on the stern is just one-half the first bearing. These inconveniences, requiring time and attention, are done away with in the method about to be described. A compass bearing is taken of some known object at any instant and the number of points contained between its direction and the ship's head, or course, are noted. A straight continuous course is then kept until the bearing of the object has altered at least 3 points, when another bearing is taken and the number of points between it and the ship's head are again noted.

18. Explanation. The manner of finding the ship's position from these data is illustrated by the following example: Assume the true bearing of an object A, Fig. 9, to be N 53° W, the true course steered to be N 3° W. After a distance of 4.5 miles is covered, according to the patent log, a second bearing of the same object is found to be S 50° W; find the ship's position.

19. By construction, its position is found as follows: Lay off from the object A, Fig. 9, the two bearings A E and AD and draw a line g h in the direction of the ship's course (= N 3° W) so that it intersects the first bearing A D (it does not matter where); on this line lay off from the intersecting point g the dis

tance run in the interval,
and from the point h thus
obtained draw a line hm
parallel to the first bear-
ing. The point x at which
this line intersects the
second bearing AE will a
be the position of the ship
at the time of the second
observation, and by draw-
ing from x a line x z
parallel to the ship's
course gh, the point of
intersection z of that line
with the first bearing AD
will show the position of

The

the ship at the instant the first bearing was taken. ship's position at both bearings is thus quickly and conveniently determined.

20. By the table on page 159, of the Nautical Tables accompanying this Course, the solution is as follows: Enter the table with the first number of points at the top and the second number of points at the side column, and take out the corresponding number; multiply this by the number of miles run in the interval. The product is the distance in miles at the time the second bearing was taken.

Referring to Fig. 9, the angle A zx between the ship's head and the first bearing is equal to 50°, or 4 points, nearly; the angle y x A at the second bearing is equal to 87° + 40° = 127°, or 114 points, nearly. Now, under 41⁄2 at the top and opposite 11 at the side column will be found the number .80;

multiplying this by the distance run in the interval, the product 4.5 X .80 = 3.6 is the distance of the ship in miles from A at the second bearing.

To find the distance A z of the ship from the object at the first bearing, enter the same table with the supplement of the second number of points at the top and the supplement of the first number of points at the side column; take out the corresponding number and multiply by the distance run. The product will be the distance of the ship from the object at the instant of the first bearing.

21. Illustrative Example. - The lighthouse at Fourteen Miles Point, Lake Superior, bore true south, the patent

log at the instant of taking the bearing registered 82 miles. Later a second bearing of the same light was observed and found to be ES E, the patent log this time registering 94 miles. The course steered between the bearings was true S W. To find the distance of the ship from the lighthouse

at the time of the second bearing, proceed as follows: Lay out the respective bearings on the chart as shown in Fig. 10. From any point a on the first bearing draw a line in the direction of the course steered S W, on this line lay off from a the distance run between the bearing, or 94 82 = 12 miles; let a b represent this distance. Through b draw the line cd parallel to the first bearing. The point x where this line. intersects the second bearing is the position of the ship at the time the second bearing was observed. The same result will be obtained by using the table referred to in Art. 20.

The angle between the first bearing and the ship's head, or course, is 4 points; the angle between the second bearing and the ship's head is 10 points; the corresponding number found in the table is .77, which, when multiplied by the distance run in the interval, or 12 miles, will give the ship's distance from the lighthouse at the second observation as 12 X .77 9.2 miles.

=

22. To Find the Distance of a Ship From the Base of a Mountain Peak or Lighthouse of Known Height. When the base of the known object is visible, its height h,

Fig. 11, above the surface of the sea may be considered as one of the perpendiculars in a right triangle, and the distance d of the ship from the object as one of the other perpendiculars, v being the angle subtended by h.