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4. How to Use the Vertical Danger Angle.-In practice at sea the student should possess Captain Lecky's “Danger Angle and Offshore 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 incorporated also in Capt. Howard Patterson's "Navigator's Pocketbook." Either of these publications may be obtained at small cost.

How the vertical danger angle is used is best shown by an illustration, as follows:

Suppose that you are about to round a cape or point of land (see Fig. 1) on which is situated a lighthouse 1, the top of which is 170 feet high. Outside of this point and within a mile of the lighthouse lies a number of rocks y

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Fig. 1 immediately below the surface of the water. These rocks cannot be seen, but their exact position is recorded on the chart. You wish to avoid the rocks by passing 1 mile outside of them, or, what is the same thing, by passing 2 miles outside of the lighthouse. In order to do this you must know what angle the lighthouse will subtend at a distance of 2 miles; in other words, you must know the

vertical danger angle for that distance. Accordingiy, you enter the tables of vertical danger angles (Lecky)—a reproduction of which is attached-with 170 feet at the top and

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2 miles in the distance column, when directly below the former and opposite the latter is found 48' 3", which is the angle the lighthouse should subtend at a distance of 2 miles. All you have to do now is to set the sextant to an angle of 48' 3" (due allowance being made for index error) and go ahead, keeping the course so that the angle will remain the same until the danger is passed. As long as the angle Isa between the top of the lighthouse I and the water-line sa is 48' 3'' you are at the correct distance; if the angle becomes less, you are outside of the 2-mile limit, as at b; if the angle becomes greater, you are nearer the rocks than desired, as at c. That such must be the case is evident from the fact that the angle lsa is greater than lba, 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 you are nearer the rocks than desired; if I drops below the water-line, the angle is smaller and you are consequently outside of your intended course.

5. 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 height of the observer's eye above the water. This error, however, will increase the angle, 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 in a safer position, unless a second danger lies outside and close to the ship's course.

6. Height of Lighthouses as Given on Different Charts.—When consulting charts for the heights of lighthouses, etc., the student should satisfy himself whether these are given in feet or meters. On American and English charts the height is invariably expressed in feet, while on charts of nearly every other nation it is expressed in meters. In the latter case, therefore, the height should be reduced to feet (1 meter = 3.28 feet) before being entered in the tables of vertical danger angles. Furthermore, it may be of importance to note that heights are measured and given for high water at ordinary spring tides.

7. Horizontal Danger Angle. -As stated before, 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. The following will illustrate the method of using the horizontal danger angle in passing a concealed danger:

Suppose that when steaming or sailing along a coast you wish to avoid some hidden rocks r, Fig. 2, by passing half a mile outside of them. On the shore there are two known objects in sight, a lighthouse 1 and a church c, both being marked on the chart. You now wish to find the danger angle corresponding to a distance of half a mile from the rocks. Proceed then as follows: With the outermost rock as center and a radius equal to half a mile, 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 cal formed by the lines a c and al. Assume it to be 52o as in the figure. This is your 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 / with your instrument in a horizontal position. When the two

objects appear in the horizon glass you are close to the circle of safety a, a ag, and when they come in contact your ship is on that circle; once on it change the course of the ship so that the two images will remain in contact until the danger is passed. As long as this is being done the ship will be on the circle of safety a, a a,, since the angles cal,

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cal, and c 1,1 are all equal, being angles in the same segment. If the angle increases, you are on the inside of the circle of safety and consequently nearer the danger than desired; if it becomes smaller, you are outside of the halfmile limit. However, by watching the angle closely and changing the course accordingly you cannot fail to keep the ship at the required distance from the rocks.

8. Comparison of the Two Methods.-Both of these methods 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. When circumstances permit the selection of a vertical and horizontal danger angle, the latter should always be preferred as being much more accurate than the former; this is on account of the horizontal angle being much larger than the vertical, and because the heights of lights and other objects on shore as recorded on charts are not always reliable and cannot be considered but approximately correct. It is evident, also, that the horizontal danger angle may be used during both night and day, whenever two lights are in sight, whereas the use of the vertical danger angle is limited to the day only.

9. How to Measure the Horizontal Angle.-For obvious reasons the student should master the method of

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(a)

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measuring horizontal angles with the sextant.

It is essentially the same as measuring vertical angles, the only difference being that the instrument is held in a horizontal position with its face either up or down, depending on which of the two objects is best suitable as a base. For instance, when measuring angles at night, if y, Fig. 3 (a), represents a fixed light and x a flash light, it is evidently more convenient to see y by reflection and x direct. Therefore, x is used as

a base and y is brought in contact with it by

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