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the log. true difference latitude, which find from tables, and mark N. or S. according as the course is northward or southward.

Apply true difference latitude (turned into degrees and minutes, if necessary) to the latitude from, and thus get latitude in. (Rule e.) Find the middle latitude. (Rule c.)

2. To find longitude in. Add together log. sin. course, log. distance, and log. secant middle latitude; the sum (rejecting 20 in the index) is the log. difference longitude, which find in tables, and mark E. or W. according as the course is eastward or westward. Apply the difference longitude (in degrees and minutes) to the longitude from, and thus get longitude in. (Rule f.)

EXAMPLE.

96. Sailed from A, S. 37° 10' W., 472-6 miles; required the latitude in and longitude in (by middle lat. method).

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Required the latitude and longitude in, by middle latitude method, in each of the following examples, having sailed

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Parallel Sailing.

In parallel sailing the ship is supposed to be kept on a parallel of latitude, as TS, fig. p. 3. The course will evidently be due east or due west, and the distance between two places as T and S, will be the arc TS between the two meridians passing through the places.

Rule V.

To find the course and distance, having given the latitude of the two places, and their longitudes.

(1.) Find the difference longitude.

(2.) The course is evidently due east or due west, according as the longitude in is to the east or west of longitude from.

(3.) To find the distance. Add together log. cos. latitude and log. difference longitude; the sum (rejecting 10 in index) is the log distance, which find in the table.

EXAMPLES.

100. Required the course and distance from A to B.

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Required the compass course and distance from A to B.

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Variation of the compass two points E, and deviation as

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Required the true course and distance from A to B, in

each of the following examples:

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Answers. Course and dist.

East 144-2'

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To find the longitude in, having given the course and distance, and latitude and longitude from.

Add together log. sec. lat. and log. distance, the sum (rejecting 10 in the index) will be the log. difference longitude. Find the natural number thereof, and turn it into

* In this example it is evident we must modify the general rule; for the diff. long. is never considered to be greater than 180°. When, therefore, the above rule gives the diff. long. greater than 180°, subtract it from 360°, and apply thereto a contrary letter to the one directed by the rule; the result will be the diff. long. to be used.

degrees, and mark it E. or W. according as the course is E. or W. Apply difference longitude to longitude from, and (Rule f.)

thus find longitude in. The latitude in is the

same as the latitude from.

EXAMPLE.

107. Sailed from A due east 1000 miles, required the latitude and longitude in.

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and latitude in lat. from = 32° 10' S.

Required the latitude and longitude in, in each of the

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The preceding rules are the principal ones used in navigation. It would be easy for the mathematical student to make for himself others, by means of the relations between the several terms course, dist., dep., &c., as shown by the fig. p. 151, in the author's volume of Astronomical Problems: he would find then no difficulty in solving problems similar to the following:

Sailed from A, in long. in 3° 10' W., 300 miles due east, and altered my longitude 10 degrees; required the latitude and longitude in.

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.. lat. in 60°, and long. in = 6° 50′ E.

Wishing to make a small island, I took the ship to windward of it in the same latitude with the island, namely, 50° 48' N. The longitude of the ship by chronometer was 20° 35′ W., and the long. of the island was 23° 50′ W. What was my distance from the island ?

In this example of parallel sailing we have given lat. 50° 48', and diff. long. 3° 15', or 195', to find distance.

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To find the place of the ship at noon, that is, its latitude and longitude, having given the latitude and longitude at the preceding noon, the compass courses, and distances run in the interval, the deviation of the compass for each course on account of local attraction, the variation of the compass, the leeway, the velocity and direction of current (if any) &c., constitutes what is called the Day's Work.

Rule VII. (the Day's Work).

(1.) Correct each course for variation (Rule g), deviation (Rule ), and leeway (Rule 7); thus get the true courses, and arrange the same in a tabular form, as in the example, p. 34. Add together the hourly distances sailed on each course, and insert the same in table opposite the true course.

(2.) Take out of the traverse table the true difference latitude and departure for each course and distance, putting

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