though it may still be used with little error. But it is better to divide the rhumb at the equator and use the middle latitude of each part separately, in connexion with the departure and the difference of longitude which correspond to that part; or we may, by an obvious extension of the principle of middle latitude sailing, take for the approximate latitude of DD' a latitude which is intermediate in amount between these two partial middle latitudes and differs less from the middle latitude which corresponds to the greater part of the rhumb, in proportion as that part is the greater. Thus, if the latitudes are l and l', taken without regard to their signs as north and south, the middle latitudes of the two parts of the rhumb are land l'; the lengths of these parts of the rhumb and the corresponding departures are proportional to l and l'; and l1, the approximate latitude of DD', is found by the proportion

The approximate latitude of DD' for all cases is expressed by the formula

in which the upper signs or the lower are to be used according as the places are on the same side of the equator or on opposite sides, and 7 and l' denote the latitudes, taken independently of their signs; for if the places are on the same side of the equator the formula becomes

NOTE. The calculations of Middle Latitude Sailing are rendered accurate by applying to the middle latitude a correction, which may be found in the table of B., p. 76 (given in the Useful Tables' after

p. 329). The method of computing this correction will be explained in the next chapter. The corrected mid. lat. is the true lat. of DD' (fig. 71) and is always a little greater than the actual mid. lat.

1. A ship sailed from Halifax (Nova Scotia) a distance of 2515 miles, upon a course S. 79° 30′ E.; find the place at which she arrived.

Ans. The place arrived at is lat. 37° 2' N., long. 9° 0′ W.; which is one mile south of Cape St. Vincent, in Portugal.

2. Find the bearing and distance of Canton from Washington.

3. A ship sails from New York a distance of 650 miles, upon S.; find the place at which she arrives.

a course S. E.

Ans. 15 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° 18′ E.

5. A ship from the Cape of Good Hope sails northwesterly, that is, between north and west, until her latitude is 22° 3' S., and her departure 3115 miles; find her course, distance sailed, longitude, and distance from Cape St. Thomas (Brazil).

6. A ship sails from Boston upon a course E. by N. until she arrives in latitude 45° 20' N.; find the distance sailed, the longitude reached, and the distance and bearing from Liverpool.

Ans. Distance sailed = 923 miles.

7. A ship sails southwesterly from Gibraltar a distance of 1500 miles, when she is in latitude 14° 43' N.; find her course, the longitude she is in, and her distance from Cape Verde.

8. A ship sails from Nantucket upon a course S. 62° 11′ E., until she has made a departure of 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 (Cape Verde Islands).

9. A ship sails southwesterly from Land's End (England) a distance of 3466 miles, when her departure is 3306 miles; find the re and the place arrived at.

Ans. The course S. 72° 30′ W.

The place arrived at is Charleston (South Carolina).

CHAPTER V.

MERCATOR'S SAILING.

41. Mercator's Sailing is an accurate method of solving those problems of rhumb sailing which involve the Difference of Longitude. [B., p. 78.]

42. Problem. To find the difference of longitude, when both latitudes and the course are known.

Solution. Let A and B (fig. 71) be the places. Suppose the rhumb AB divided into very small portions A a, a b, bc, &c., which are such that the difference of longitude is the same for each of them. Let

D= the required difference of longitude of B and A,

d = the small difference of longitude which corresponds to either of the small portions of the rhumb,

L

the given latitude of B,

L' the given latitude of A,

7 the latitude of any one of the points of division, as c,

=

l' = the latitude of b, the next point towards A,

C= the given course,

n the number of portions into which BA is divided.

Now, since we suppose the rhumb to be divided into as many parts as we please, we may suppose each of the parts to be so small that the formulas of middle latitude sailing can be applied to it without error; so that we have for any one of them, as c b, by (232), d=(-1) tang. CX sec. (1' + 7),

or, by dividing by 2 tang. C, we have, by (6),

(239)

which, substituted in (240) multiplied by sin. 1', gives

and (243) may be written in the usual form of a proportion sin. (-7): cos. (l'+1)=m:1;

(244)

But if in (47), in which A and B may have any values, we take

Now, since the course C is everywhere the same, and since dis assumed to be the same for each portion of the rhumb, m is, by (243), the same for each portion of the rhumb, and, therefore, by (249), M, the ratio of cotan. (45°-7') to cotan. (45°-7), is likewise the same for each portion of the rhumb. Hence the successive values of cotan. (45°-7), for the points B, . . . . c, b, a, A, form a geometric progression, of which

cotan. (45°-L) the first term,

cotan. (45° —L') = the last term,

M= the common ratio,

Therefore, by the theory of geometric progression,

cotan. (45° — L') = cotan. (45° — § L) . M",

and, by logarithms,

(251)

log. cotan. (45° — § L') —log. cotan. (45° — † L) = log. M". (252)

cotan. (45° 7)

;

(248)

(249)

-

m

=M.

(250)