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3. A ship sails from New York a distance of 675 miles, upon a course S. E. 1 S.; find the place at which it arrives.

Ans. Three miles to the west of Georgetown in Bermuda.

4. Find the bearing and distance of Portland (Maine) from New Orleans.

Ans. The bearing N. 49° 24′ E.

The distance 1257 miles.

5. A ship from the Cape of Good Hope sails northwesterly until its latitude is 22° 3' S., and its departure 3110 miles; find its course, distance sailed, longitude, and its distance from Cape St. Thomas (Brazil).

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Distance to the Cape St. Thomas = 22 miles.

6. A ship sails from Boston upon a course E. by N. until it arrives in latitude 45° 21′ N.; find the distance, its longitude, and its distance and bearing from Liverpool.

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7. A ship sails southwesterly from Gibraltar a distance of 1500 miles, when it is in latitude 14° 44' N.; find its course and longitude, and distance from Cape Verde.

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8. A ship sails from Nantucket upon a course S. 62° 11′ E., until its departure is 2274 miles; find the distance sailed, and the place arrived at.

Ans. Distance 2571 miles.

The place arrived at is 261 miles north of Santa Cruz.

9. A ship sails southwesterly from Land's End (England) a distance of 3461 miles, when its departure is 3300 miles; find the course and the place arrived at.

Ans. The course = S. 72° 27' W.

The place arrived at is Charleston (South Carolina).

CHAPTER V.

MERCATOR'S SAILING.

41. THE object of Mercator's Sailing is to give an accurate method of calculating the difference of longitude. [B. p. 78.]

42. Problem. To find the difference of longitude, when the distance, the course, and one latitude are known.

Solution. Let AB (fig. 16) be the ship's track. Divide it into the small portions A a, ab, bc, &c., which are such that the difference of longitude is the same for each of them, and let

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the latitude of one of the points of division as b,

the latitude of the next point c,

C the course.

The distance be may then be supposed so small, that the formulas of middle latitude sailing may be applied to it; and (232) gives

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But, (7

7) is a small arc expressed in minutes, and by (14)

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and (243) may be written in the usual form of a proportion

sin. (7—7): cos. 1 (7' + 1) = m : 1

(244)

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Hence, the successive values of cotan. (45°-7) at the points

A, a, b, &c., form a geometric progression; and if

D= the difference of longitude of A and B,

ก the number of portions of AB;

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cotan. (45° — 1 L') — cotan. (45° — 1 L) M2,

(252)

and by logarithms

log. cotan. (45° — 1 L') — log. cotan. (45° — 1 L) = log. M”. (253)

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which, substituted in (253), gives by a simple reduction

(254)

(255)

(256)

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cosec.1'
log. e
× tang. C = D.
cosec. 1'
log. e

log. cotan. (45° — ÷ L)]

(257)

Now the value of log. cotan. (45°-L) has been calculated for every mile of latitude, and inserted in tables. [B. Table III.] It is called the Meridional Parts of the Latitude, and the method of computing it is given in the following section.

The difference between the meridional parts of the two latitudes, when the latitudes are both north or both south, is called the Meridional Difference of Latitude; but when one of the latitudes is north and the other south, the sum of the meridional parts is the meridional difference of latitude.

Hence (257) gives

D= diff. long. mer. diff. lat. X tang. course.

(258)

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