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The course is 3 points, or 33° 45' (Art. 35.)
Sin. 33° 45': 21.1=Depart.

R: 38:

Cos. 33° 45'

31.6-Diff. Lat.

Example 2.

A ship sails S. 29° E., 34 leagues. Her departure and difference of latitude are required.

Ans. 16.5 and 29.7 leagues. The proportions in this and the following cases may be varied, by making different sides radius, as in Trigonometry, Sec. III.

The course

CASE II.

45. Given And departure; to find

The distance, and
Difference of lat.

Making the distance radius, (Trig. 137.)

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A ship leaving a port in latitude 42° N. has sailed S. 37° W. till she finds her departure 62 miles. What distance has and in what latitude has she arrived ?

she run,

Sin. 37° 62: :

Rad. 103-Distance.

Cos. 370: 82.3-Diff. of latitude.

The difference of latitude is 82.3 miles, or 1° 22′.3. (Art. 41.) This is to be subtracted from the original latitude of the ship, because her course was towards the equator. The remainder is 40° 47'.7, the latitude on which she has arrived.

Example 2.

A ship leaves a port in latitude 63° S., and runs N. 54° E. till she makes a harbor where her departure is found to be 74 miles; how great is the distance of the two places, and what is the latitude of the latter?

The distance is 91 miles; and the latitude of the latter place is 62° 06'.2.

CASE III.

46. Given

{

The course, and
Diff. of latitude;

}

to find {

The distance,
And departure.

Making the distance radius,

Cos. Course: Diff. Lat. : S Rad.

Sin. Course Departure.

{

Distance.

Example.

A ship sails S. 50° E. from latitude 7° N., to latitude 4° S. Required her distance and departure.

As the two latitudes are on different sides of the equator, the distance of the parallels is evidently equal to the sum of the given latitudes. This is 11°, or 660 miles. The distance is 1026.8 miles, and the departure 786.

47. Given

CASE IV.

}

to find

The course, and

Diff. of latitude.

The distance,

And departure;
Making the distance radius, (Trig. 135.)
Dist. Rad. Depart. Sin. Course,
Dist.:: Cos. Course: Diff. lat.

Rad.

Example.

A ship having left a port in Lat. 3° N., and sailing between S. and E. 400 miles, finds her departure 180 miles. What course has she steered, and what is her latitude?

Her latitude is 2° 57' S., and her course S. 26° 44′ E.

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A vessel sails between N. and E. 66 miles, from Lat. 34° 50' to Lat. 35° 40'. Required her course and departure. The course is N. 40° 45′ E. and the departure 43.08 miles.

49. Given

{

CASE VI.

The course,

The departure, and
Diff. of latitude;

}

to find and distance.

Making the difference of latitude radius, (Trig. 139.) .
Diff. Lat. Rad.:: Depart.: Tan. Course,

Rad. Diff. Lat. :: Sec. Course: Distance.

Example.

A ship sails from the equator between S. and W., till her

latitude is 5° 52', and her departure 264 miles. Required her course and distance.

The course is S. 36°52′ W., and the distance is 440 miles.

Examples for practice.

1. Given a ship's course S. 46° E., and departure 59 miles; to find the distance and difference of latitude.

2. Given the distance 68 miles, and departure 47; to find the course and difference of latitude.

3. Given the course SSE., and the distance 57 leagues; to find the departure and difference of latitude.

4. Given the course NW. by N., and the difference of latitude 2° 36'; to find the distance and departure.

5. Given the departure 92, and the difference of latitude 86; to find the course and distance.

6. Given the distance 123, and the difference of latitude 96; to find the course and departure.

THE TRAVERSE TABLE.

50. To save the labor of calculation, tables have been prepared, in which are given the departure and difference of latitude, for every degree of the quadrant, or for every quarter of a degree. These are called Traverse tables, or tables of Departure and Latitude. The distance is placed in the left hand column, the departure and difference of latitude directly opposite, and the degrees, if less than 45° or 4 points, at the top of the page, but if more than 45°, at the bottom. titles at the top of the columns correspond to the courses at the top; and the titles at the bottom, to the courses at the bottom; the difference of latitude for a course greater than 45°, being the same as the departure for one which is as much less than 45°. See Trig. 104.

The

If the given distance is greater than any contained in the table, it may be divided into parts, and the departure and difference of latitude found for each of the parts. The sums of the numbers thus found will be the numbers required.

The departure and difference of latitude for decimal parts may be found in the same manner as for whole numbers, by supposing the decimal point in each of the columns to be moved to the left, as the case requires.

With the aid of a traverse table, all the cases of plane sailing may be easily solved by inspection.

Ex. 1. Given the course 33° 45'; and the distance 38 miles; to find the departure and difference of latitude.

Under 3330, and opposite 38, will be found the difference of latitude 31.6, and the departure 21.11; the same as in page 21.

2. Given the course 57°, and the distance 163.

The departure and diff. of lat. for 100 are 83.87 and 54.46

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3. Given the course 39°, and the distance 18.23. The departure and diff. of lat. for 18. are 11.33 and 13.99

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4. Given the course 41° 15', and the departure 60. Under 4110, and against the departure 60, will be found the difference of latitude 68.42, and the distance 91.

5. Given the distance 63, and the departure 56.

Opposite the distance 63, find the departure 56; in the adjoining column will be the latitude 28.85, and at the bottom, the course 6230.

6. Given the departure 72, and the difference of latitude 37.

Opposite these numbers in the columns of latitude and departure, will be found the distance 81, and at the foot of the columns, the course 623o.

51. The traverse table is useful, not only for taking out departure and difference of latitude; but for finding by inspection the sides and angles of any right angled triangle whatever. In plane sailing, the distance is the hypothenuse, (see Fig. 20.) the difference of latitude is the perpendicular, the departure is the base, and the course is the acute angle

at the perpendicular. If then the hypothenuse of any rightangled triangle whatever, be found in the column of distances, in the traverse table; the perpendicular will be opposite in the latitude column, and the base in the departure column; the angle at the perpendicular, being at the top or bottom of the page.

Ex. 1. Given the hypothenuse 24, and the angle at the perpendicular 544°; to find the base and perpendicular by inspection.

Opposite 24 in the distance column, and over 540 will be found the base 19.54 in the departure column, and the perpendicular 13.94 in the latitude column.

2. Given the angle at the perpendicular 3710, and the base 46; to find the hypothenuse and perpendicular.

Under 3710, look for 46 in the departure column; and opposite this will be found the perpendicular 60.5 in the latitude column, and the hypothenuse 76 in the distance col

umn.

3. Given the perpendicular 36, and the base 30.21; to find the hypothenuse and angles.

Look in the columns of latitude and departure, till the numbers 36 and 30.21 are found opposite each other; these will give the hypothenuse 47. and the angle at the perpendicular 40°.

SECTION II.

PARALLEL AND MIDDLE LATITUDE SAILING.

52. By the methods of calculation in plane sailing, a ship's course, distance, departure, and difference of latitude are found. There is one other particular which it is very important to determine, the difference of longitude. The departure gives the distance between two meridians in miles. But the situations of places on the earth, are known from their latitudes and longitudes; and these are measured in degrees. The lines of longitude, as they are drawn on the globe, are

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