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2. A ship sails in the N.W. quarter, 248 miles, till her departure is 135 miles, and her difference of longitude 310 miles: required her course, the latitude left, and the latitude come to.

Course N. 32° 59′ W.; lat. left 62° 27′ N.; lat. in 65° 55′ N.

3. A ship, from latitude 37° N., longitude 9° 2′ W., having sailed between the N. and W., 1027 miles, reckons that she has made 564 miles of departure; what was her direct course, and the latitude and longitude reached?

Course N. 33° 19' W. or N. W. by N. nearly; lat. 51° 18′ N.; long. 22°8′ W.

4. Required the course and distance from the east point of St. Michael's, lat. 37° 48′ N., long. 25° 13′ W., to the Start Point, lat. 50° 13′ N., long. 3° 38′, the middle latitude being corrected by Workman's Table.

Course N. 51° 11' E.; distance 1189 miles.

Mercator's Sailing.

(68.) It has been already seen that when a ship sails on any oblique rhumb the difference of latitude, the departure, and the distance run, are truly represented by the sides of a right-angled plane ci

B

G

triangle. The departure B'B represents the sum of all g the very small meridian distances, or elementary departures, b'b, c'c, &c. in the diagram, at page 124, the difference of latitude AB represents the sum of all the corresponding small difference in the figure referred to; and the distance AB, the sum of all the distances to which A these several departures and differences belong, and each of these elements are supposed to be taken so excessively small as to form on the sphere a series of triangles, differing insensibly from plane triangles.

Let Ab'b in the annexed diagram represent one of these elementary triangles, b'b will be one of the elements of the departure, and Ab, the corresponding difference of latitude; and as b'b is a small portion of a parallel of latitude, it will be to a similar portion of the equator, or of the meridian, as the cosine of its latitude to radius (66). This similar portion of the equator, or of the meridian, being the difference of

have D=

longitude between b' and b. Suppose now the distance Ab prolonged to p, till the departure p'p is equal to the difference of longitude of b′, and b, then b'b will be to p'p as the cosine of the latitude of bb to the radius; but b'b: p'p :: Ab′ : Ap'; hence the proper difference of latitude Ab' is to the increased difference Ap' as the cosine of the latitude of b'b to the radius. Calling, therefore, the proper difference of latitude d, the increased difference D, the latitude of b'b, C, and the radius R, we Rd = Rd sec. l; the ship, therefore, having made the cos. / small departure b'b, and the difference of latitude Ab, must continue her course till the difference of latitude becomes D, in order that her departure may become equal to the difference of longitude corresponding to b'b. Conceiving all the elementary distances to be in this manner increased, the sum of all the corresponding increased departures will necessarily be the whole difference of longitude made by the ship during the course; to represent, therefore, the difference of longitude due to the departure B′B, and difference of latitude AB', we must prolong AB′ till AC′ is equal to the sum of all the elementary differences increased as above, and the departure C'C, due to this difference of latitude, will represent the difference of longitude actually made in sailing from A to B. The determination of AC' requires the previous determination of all its elementary parts; if d be taken equal to 1′ each of these parts will be expressed by D=1′ sec. 1, from which equation the values of D, corresponding to every minute of l, from the equator to the pole, may be calculated; and by the continued addition of these there will be obtained, in succession, the values of the increased latitude corresponding to 1', 2′, 3′, &c. of proper latitude; these values are called the meridional parts, corresponding to the several proper latitudes, and when registered in a table form a table of meridional parts, given in all books on Navigation.

The following may serve as a specimen of the manner in which such a table may be constructed, and, indeed, of the manner in which the first table of meridional parts was actually formed by Mr. Wright, the proposer of this ingenious and valuable method.

Mer. pts. of 1'= nat. sec. l'.

Mer. pts. of 2′ = nat. sec. 1' + nat. sec. 2′.

Mer. pts. of 3′ = nat. sec. l'+nat. sec. 2′+nat. sec. 3'.

Mer. pts. of 4′ = nat. sec. 1' + nat. sec. 2′+ nat. sec. 3′+ nat. sec. 4'.

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Hence, by means of a table of natural secants, we have

Mer. pts. of l' =

Nat. secs. Mer. parts.

1.0000000= 1.0000000

Mer. pts. of 2' = 1·0000000 + 1·0000002 = 2·0000002

*

Mer. pts. of 3′ = 2·0000002 + 1·00000043·0000006

Mer. pts. of 4' = 3·0000006 + 1·0000007 = 4·0000013

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There are other methods of construction, but this is the most simple and obvious; we shall, however, presently have to advert to another process of computation, by which the meridional parts for any latitude may be found independently of previous calculations. The meridional parts, thus determined, are all expressed in geographical miles, because in the general expression D= 1′ sec. 7, 1' is a geographical mile.

Having thus formed a table of meridional parts, (see Riddle's Navigation, or Robertson's Treatise,) if we enter it with the latitudes sailed from, and come to, and take the difference of the corresponding parts in the table, the remainder will be the meridional difference of latitude, or the line AC' in the preceding diagram, and the difference of longitude C'C will then be obtained by this proportion, viz.

1. As radius is to the tangent of the course, so is the meridional difference of latitude to the difference of longitude; or if the departure be given instead of the course then the proportion will be

2. As the proper difference of latitude is to the departure, so is the meridional difference of latitude to the tangent of the course. Other proportions immediately suggest themselves from the preceding figure. (69.) As an example of Mercator's, or more properly of Wright's, sailing, let us take the following.

1. Required the course and distance from the east point of St. Michael's to the Start Point.

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(70.) In the absence of a table of meridional parts, a table of logarithmic tangents may be employed for the same purposes; and, indeed, the meridional parts corresponding to any given latitude may be expeditiously computed by help of such a table, and independently of any previous computations.

This method of computation will be found investigated at note C, at the end, in which we have mentioned one or two additional particulars respecting Mr. Wright's method. The practical rule is as follows, viz.

If the log. tangent of half the complement of any latitude be subtracted from 10, and the remainder be multiplied by 7915-7044679, &c. the product will give the meridional parts in miles, corresponding to that latitude.

From this rule the method of operating with logarithmic tangents, instead of with meridional parts, may be easily derived. Call t, t', the logarithmic tangents of the half complements of the latitudes left and reached, and put a for the constant multiplier 7915·7044, &c. Then, by the rule just given, the meridional difference of latitude will be

a {(10 — ť′) — (10 — t) } = a (t — t'′) = (t — t′) 10000 ÷

10000

a

Now log.

10000

a

1015104, therefore, the logarithm of the me

ridional difference of latitude is found by removing the decimal point in the difference t -t' four places to the right, and then subtracting the constant number 1015103. Hence, if instead of the logarithm of the radius 10, we use 10'1015104, and instead of the meridional parts the logarithmic tangents t, t', of the complements of the half latitudes, taking care in setting down the difference of these to remove the decimal point four places to the right, the proportion (1), at page 129, may be still employed. Thus, taking the foregoing example, the operation, by this method will be as follows.

St. Michael's 26°6′ .. t 9.6901030

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The reason why the resulting logarithm here does not exactly coincide with that obtained by using the meridional parts, is that the meridional parts have been computed to but one place of decimals; if they had been computed to two or three places, the two results would have been exactly the same.

2. Given the Lizard in lat. 49° 55′ N. Barbadoes in lat. 13° 10' N., and their difference of longitude 53°, or 3180′ W.; to determine the course and distance.

Course S. 49° 59′ W.; distance 3429 miles.

3. A ship sails from lat. 37° N. long., 22° 56′ W, on the course N., 33° 19′ E., till she arrives at lat. 51° 18′ N.: required the distance sailed, and the longitude arrived at.

Distance 1027 miles; longitude in 9°45′ W.

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