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farthest from each other at the equator, and gradually converge towards the poles. A ship, in making a hundred miles of departure, may change her longitude in one case 2 degrees, in another 10, and in another 20. It is important, then, to be able to convert departure into difference of longitude; that is, to determine how many degrees of longitude answer to any given number of miles, on any parallel of latitude. This is easily done by the following

THEOREM.

53. AS THE COSINE OR LATITUDE,

TO RADIUS;

SO IS THE DEPARTURE,

TO THE DIFFERENCE OF LONGITUDE.

By this is to be understood, that the cosine of the latitude is to radius; as the distance between two meridians measured on the given parallel, to the distance between the same meridians measured on the equator.

Let P (Fig. 21.) be the pole of the earth, A a point at the equator, L a place whose latitude is given, and LO a line perpendicular to PC. Then CL or CA is a semi-diameter of the earth, which may be assumed as the radius of the tables; PL is the complement of the latitude, and OL the sine of PL, that is, the cosine of the latitude.

If the whole be now supposed to revolve about PC as an axis, the radius CA will describe the equator, and OL the given parallel of latitude. The circumferences of these circles are as their semi-diameters OL and CA, (Sup. Euc. 8. 1.) And this is the ratio which any portion of one circumference has to a like portion of the other. Therefore OL is to CA, that is, the cosine of latitude is to radius, as the distance between two meridians measured on the given parallel, to the distance between the same meridians measured on the equator.

Cor. 1. Like portions of different parallels of latitude are to each other, as the cosines of the latitudes.

Cor. 2. A degree of longitude is commonly measured on the equator. But if it be considered as measured on a parallel of latitude, the length of the degree will be as the cosine of the latitude.

The following table contains the length of a degree of longitude for each degree of latitude.

d.l. miles. [d.1.| miles. ||d.l.| miles. d.l.] miles. ||d.l. miles. d.l. miles. 159.99 16 57.67 31 51.43 46 41.68 61 29.09 76 14.52 259.96 17 57.38 32 50.88 47 40.92 62 28.17 77 13.50 359.92 18 57.06 33 50.32 48 40.15 63 27.24 78 12.47 4 59.85 19 56.73 34 49.74 4939.36 64 26.30 79 11.45 559.77 20 56.38 35 49.15 50 38.57 65 25.36 80 10.42 659.67 2156.01 36 48.54 5137.76 66 24.40 81 9.39 759.55 22 55.63 37 47.92 52 36.94 67 23.44 82 8.35 859.42 23 55.23 38 47.28 5336.11 68 22.48 83 7.31 9 59.26 24 54.81 39 46.63 54 35.27 69 21.50 84 6.27 10 59.08 25 54.38 40 45.96 5534.41 70 20.52 85 5.23 11 58.89 26 53.93 41 45.28 56 33.55 71 19.53 86 4.19 12 58.68 27 53.46 42 44.59 57 32.68 72 18.54 87 3.14 13 58.46 28 52.97 43 43.88 5831.80 73 17.54 88 2.09 14 58.22 29 52.47 44 43.16 5930.90 74 16.54 89 1.05 15 57.95 3051.96 45 42.436030.007515.53 90 0.00

The length of a degree of longitude in different parallels is also shown by the Line of Longitude, placed over or under the line of chords, on the plane scale. (See Trig. 165.)

54. The sailing of a ship on a parallel of latitude* is called Parallel Sailing. In this case, the departure is equal to the distance. The difference of longitude may be found by the preceding theorem; or if the difference of longitude be given, the departure may be found by inverting the terms of the proportion. (Alg. 380. 3.)

55. The Geometrical Construction is very simple. Make CBD (Fig. 22.) a right angle, draw BC equal to the departure in miles, lay off the angle at C equal to the latitude in degrees, and draw the hypothenuse CD for the difference of longitude. The angle C, and the sides BC and CD, of this triangle, have the same relations to each other, as the latitude, departure, and difference of longitude.

For Cos. C: BC::R: CD (Trig. 121.)

And Cos. Lat. Depart.:: R: Diff. Lon. (Art. 53.)

See Note C.

56. The parts of the triangle may be found by inspection in the traverse table. (Art. 51.) The angle opposite the departure is D the complement of the latitude, and the difference of longitude is the hypothenuse CD. If then the departure be found in the departure column under or over the given number of degrees in the co-latitude, the difference of longitude will be opposite in the distance column.

Example I.

A ship leaving a port in Lat 38° N. Lon. 16° E. sails west on a parallel of latitude 117 miles in 24 hours. What is her longitude at the end of this time?

:

:

Cos. 38° Rad. 117: 148-2° 28 the difference of longitude.

This subtracted from 16° leaves 13° 31' the longitude required.

Example II.

What is the distance of two places in Lat 46° N. if the longitude of the one is 2° 13' W. and that of the other 1° 17/ E.?

As the two places are on opposite sides of the first meridian, the difference of longitude is 2° 13'+1° 17'-3° 30', or 210 minutes. Then

Rad.

Cos. 46° 210 145.83 miles, the departure, or the distance between the two places.

Example III.

A ship having sailed on a parallel of latitude 138 miles, finds her difference of longitude 4° 3' or 243 minutes. What is her latitude?

Diff. Lon. 243: Dep. 138 :: Rad. : Cos. Lat. 55° 23'.

Example IV.

On what part of the earth are the degrees of longitude half as long as at the equator? Ans. In latitude 60.

2.4

2

MIDDLE LATITUDE SAILING.

57. By the method just explained, is calculated the difference of longitude of a ship sailing on a parallel of latitude. But instances of this mode of sailing are comparatively few. It is necessary then to be able to calculate the longitude when the course is oblique. If a ship sail from A to C, (Fig. 18.) the departure is equal to om+sn+tC. But the sum of these small lines is less than BC, and greater than AD. (Art. 40.) The departure, then, is the meridian distance measured not on the parallel from which the ship sailed, nor on that upon which she has arrived, but upon one which is between the two. If the exact situation of this intermediate parallel could be determined, by a process sufficiently simple for common practice, the difference of longitude would be easily obtained. The parallel usually taken for this purpose, is an arithmetical mean between the two extreme latitudes. This is called the Middle Latitude. The meridian distance on this parallel is not exactly equal to the departure. But for small distances, the error is not material, except in high latitudes.

The middle latitude is equal to half the sum of the two extreme latitudes, if they are both north or both south: but to half their difference, if one is north and the other south.

58. In middle latitude sailing, all the calculations are made in the same way as in plane sailing, excepting the proportions in which the difference of longitude is one of the terms. The departure is derived from the difference of longitude, and the difference of longitude from the departure, in the same manner as in parallel sailing, (Arts. 53, 54.) only substituting in the theorem the term middle latitude for latitude.

THEOREM. I.

AS THE COSINE OF MIDDLE LATITUDE,

TO RADIUS;

SO IS THE DEPARTURE,

TO THE DIFFERENCE OF LONGITUDE.

59. The learner will be very much assisted in stating the proportions, by keeping the geometrical construction steadily in his mind. In fig. 20 we have the lines and angles in plane sailing, and in Fig. 22, those in parallel sailing.

By

bringing these together, as in Fig. 23, we have all the parts in middle latitude sailing. The two right angled triangles, being united at the common side BC, which is the departure, form the oblique angled triangle ACD.

60. The angle at D is the complement of the middle latitude. (Art. 55.) Then in the triangle ACD, (Trig. 143.) Sin. D AC:: Sin. A: DC; that is,

THEOREM II.

AS THE COSINE OF MIDDLE LATITUDE,
TO THE DISTANCE;

SO IS THE SINE OF THE COURSE,

TO THE DIFference of longitude.

61. The two preceding theorems, with the proportions in plane sailing, are sufficient for solving all the cases in middle latitude sailing. A third may be added, for the sake of reducing two proportions to one.

In the triangle BCD (Fig. 23.) Cos. BCD: R:: BC: CD And in the triangle ABC, AB: R:: BC: Tan. A. The means being the same in these two proportions, the extremes are reciprocally proportional. (Alg. 387.) We have then

Cos. BCD AB:: Tan. A: CD; that is,

THEOREM III.

AS THE COSINE OF MIDDLE LATITUDE,
TO THE DIFFERENCE OF LATITUDE;
SO IS THE TANGENT OF THE COURSE,

TO THE DIFFERENCE OF LONGITUDE.

Among the other data in middle latitude sailing, one of the extreme latitudes must always be given.

Example I.

At what distance, and in what direction, is Montock Point from Martha's Vineyard; the former being in Lat. 41° 04' N. Lon. 72° W., and the latter in Lat. 41° 17' N. Lon. 70° 48' W.?

Here are given the two latitudes and longitudes, to find the course and distance.

The difference of longitude is 72/
The difference of latitude

The middle latitude

13/

41° 10'

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