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THE DANGER ANGLE

28. The danger angle, which may be either vertical or horizontal, is the name given to a method that is used when coasting to avoid hidden dangers, such as rocks, shoals, sunken derelicts, or other obstructions situated immediately at or below the surface of the water. By its use, any such dangerous obstacle may be passed or rounded at any desired distance. For this reason its proper name should be safety angle instead of danger angle, although the latter term is the one most commonly used.

29. The vertical danger angle is based on the known height of a lighthouse, or other known object the height of which is given on the chart or in the List; and the distance corresponding to such angle as is subtended by this height.

30. The horizontal danger angle is based on the angular distance or the angle subtended by two knowi objects given in the chart and visible at the same time; it is an application of the geometrical properties of the circle, viz., that angles in the same segment of a circle are equal.

31. Application of the Vertical Danger Angle. The student should possess Captain Lecky's “Danger Angle and Off-Shore Distance Tables," in which are given the sextant, or danger, angle for heights up to 1,100 feet and the corresponding distance expressed in miles and tenths of a nautical mile. A useful set of tables of vertical danger angles is also incorporated in Captain H. Patterson's "Navigators’ Pocketbook." Either of these publications may be obtained at small cost. The following explanation will show how the vertical danger angle is used when passing a concealed danger.

32. Explanation. - Assume a steamer to be about to round a point of land (see Fig. 14) on which a lighthouse / 170 feet high is situated. Outside of this point, and within a mile of the lighthouse, lie a number of rocks r immediately below the surface of the water. They cannot be seen, but their position is indicated on the chart. It is desired to avoid these rocks by passing 1 mile outside of them or, what amounts to the same, by passing 2 miles outside of the lighthouse. In order to do this it is necessary to know what angle the lighthouse will subtend at a distance of 2 miles; in

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other words, the vertical danger angle for that distance must be known. Accordingly, we enter the Tables of Vertical Danger Angles (Lecky), a reproduction of which is shown below,

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with 170 feet at the top and 2 miles in the distance column, when directly below the former and opposite the latter is found 48' 3''. This is the angle that the lighthouse should subtend at a distance of 2 miles. All that has to be done now is to set the sextant to an angle of 48' 3" (due allowance being given for index error) and to go ahead, keeping the course so that the angle will remain the same until the danger is passed. As long as the angle Is a between the top of the lighthouse l and the water-line sa is 48' 3'', the ship is at the correct distance; if the angle becomes less, she is outside of the 2-mile limit, as at b; if the angle becomes greater, she is nearer the rocks than desired, as at c. That such must be the case is evident from the fact that the angle Isa is greater than l b a but smaller than lca, and we know that the greater the angle becomes the shorter will be the distance. It is not necessary to move the index bar of the sextant at all, simply have it clamped at the required angle, for, if I rises above the water-line sa, as seen in the horizon glass of the instrument, the angle is larger than that set and means that the ship is nearer the rocks than is desirable; if I drops below the water-line, the angle is smaller, and the ship is consequently outside of her intended course.

33. In observing vertical danger angles, it is advisable that the observer be as near to the surface of the water as possible; this will tend to minimize the error caused by the elevation of the observer's eye above the water-line. This errror, however, will increase the angle subtended by the object, and, since a greater angle corresponds to a shorter distance, it is evident that the ship will actually be farther away from the danger than the recorded distance and thus be in a safer position, unless a second danger lies outside and close to the ship's course.

This method may prove useful in cases where buoys and marks indicating the location of shoals and rocks have been destroyed or carried away by ice or otherwise.

34. Illustrative Example. - Assume that the 30-foot spar buoy that marks the northern end of the rocky shoal extending out from Big Bay Point, Mich., has been carried away by ice.

The end of the shoal is situated about 1.8

statute miles NW from the lighthouse at Big Bay Point. The top of this lighthouse is 108 feet above the lake level. You are bound west and wish to keep clear of the rocks by passing 1 statute mile outside of them, or at a distance of 2.8 statute miles ( = 2.5 nautical miles) from the lighthouse. To find the vertical danger angle to be used in this case, proceed as follows: Examining the Tables of Vertical Danger Angles referred to, we find, against 110 feet at the top and 2.5 miles in the side column, the angle 24' 53". Hence, in order to clear the rocks at the desired distance, the angle subtended by the top of the lighthouse and its base (lake level) should be nearly 25'; therefore, by setting the sextant to that angle and altering the course of the ship so that this angle will remain constant, the rocks may be passed with tolerable accuracy at the distance required.

35. Application of the Horizontal Danger Angle. As previously stated, the horizontal danger angle is based on the angular distance between two known visible objects. This method is very valuable and much to be preferred to the vertical danger angle. It is used in the following manner.

36. Explanation. -Suppose that when steaming along a coast it is decided to avoid some hidden rocks R, Fig. 15, by passing mile outside of them. On the shore there are two known objects in sight, a lighthouse L and a church C, both being marked on the chart. Then, to find the danger angle corresponding to a distance of half a mile from the rocks, proceed as follows: With the outermost rock as a center and a radius equal to } mile (either statute or nautical) describe a circle on the chart. Then, through the most seaward point a of this circle and the points C and L, describe another circle; connect a with C and L; measure with a protractor the angle Ca L formed by the lines a C and a L. Assume it to be 52°, as in the figure. This is the required horizontal danger angle. Now, set that angle on the sextant (neglecting the index error if it is small) and watch the two selected objects C and L, holding the instrument in a horizontal position. When the two objects appear in the horizon glass, the ship is close to the circle of safety a, a an, and when they come in contact, the ship is on that circle; once on the circle, change the course of the ship so that the two images will remain in contact until the danger

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is passed. As long as this is being done the ship will be on the circle of safety a, a az, since the angles Ca, L, Ca L, and Ca, L are all equal, being angles in the same segment. If the angle increases, the ship is on the inside of the circle of safety and consequently nearer the danger than is desirable; if it becomes smaller, the ship is outside of the halfmile limit. However, by watching the angle closely and

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