Formula (1) above may be expressed by the following rule. Divide the distance sailed by the cosine of the latitude, and the quotient will be the difference of longitude.

EXAMPLES.

1. A ship from latitude 53° 56' N., longitude 10° 18′ E., has sailed due west, 236 miles : required her present longitude.

By the rule

2. If a ship sail E. 126 miles, from the North Cape, in lat. 70° 10' N., and then due N., till she reaches lat. 73° 26' N.; how far must she sail W. to reach the meridian of the North Cape?

Here the ship sails on two parallels of latitude, first on the parallel of 71° 10', and then on the parallel of 73° 26', and makes the same difference of longitude on each parallel. Hence by the corollary,

3. A ship in latitude 32° N. sails due east, till her difference of longitude is 384 miles; required the distance run.

325.6 miles.

4. If two ships in latitude 44° 30′ N., distant from each other 216 miles, should both sail directly south till their distance is 256 miles, what latitude would they arrive at?

32° 17' N.

5. Two ships in the parallel of 47° 54′ N., have 9° 35′ difference of longitude, and they both sail directly south, a distance of 836 miles :

required their distance from each other at the parallel left, and at that reached.

100. Having seen how the longitude which a ship makes when sailing on a parallel of latitude may be determined, we come now to examine the more general problem, viz., to find the longitude a ship makes when sailing upon any oblique rhumb.

There are two methods of solving this problem, the one by what is called middle latitude sailing, and the other by Mercator's sailing. The first of these methods is confined in its application, and is moreover somewhat inaccurate even where applicable; the second is perfectly general, and rigorously true; but still there are cases in which it is advisable to employ the method of middle latitude sailing, in preference to that of Mercator's sailing; it is, therefore, proper that middle latitude sailing should be explained, especially since, by means of a correction to be hereafter noticed, the usual inaccuracy of this method may be rectified.

very inaccurate when the course is small, and the distance run great; for it is plain that the middle latitude distance between the extreme meridians will be much greater than the departure, if the track A B cuts the successive meridians at a very small angle.

The principle approaches nearer to accuracy as the angle A of the course increases, because then as but little advance is made in latitude, the several component departures lie more in the immediate vicinity of the middle latitude parallel. But since in very high latitudes, a small advance in latitude makes a considerable difference in meridional distance, this principle is not to be recommended in such latitudes if much accuracy is required.

By means, however, of a small table of corrections, constructed by Mr. WORKMAN, the imperfections of the middle latitude method may be removed, and the result of it rendered in all cases accurate. This table we have given at the end of the present volume.

The rules for middle latitude sailing may be thus deduced.

diflang

dap.

dist

It has been seen at (Art. 97), that the difference of latitude, departure, and distance sailed on any oblique rhumb, may be all accurately represented by the sides AB', B'B, AB, of a right angled plane triangle. Now, by the present hypothesis, the departure B'B is equal to the middle latitude distance between the meridians of the places sailed from, and arrived at, so that the difference of longitude of the two places of the ship is the same as if it had sailed the distance B'в on the middle latitude parallel; the determination of this difference of longitude is, therefore, reduced to a case of parallel sailing; and since, as we have seen (p. 215), the formula for parallel sailing is a proportion which expresses the relation between the elements of a right angled plane triangle in which the base is the dist. sailed, the angle at the base the lat., and the hypothenuse the diff. of long., let B'BA' be this triangle, in which, according to the theory of mid. lat. sailing, the departure B'в takes the place of the dist. sailed. From these triangles, the two partial ones of which are right angled, and the total one not, we have the following theorems, viz., in the triangle A'B'B,

that is,

COS A'BB': BB':: radius: BA'

Ꭺ

I. Cos. mid. lat. : departure :: radius: diff. of long.

In the triangle A'BA, which is not right angled,

that is,

sin A' AB sin A: A'B;

II. Cos mid. lat. : distance: sin course: diff. long.

In the triangle ABB', we have the proportion (Art. 41),

R: tan A:: AB' :: BB'

comparing this with the first proportion above, observing that the extremes

of this are the means of that, we have

that is,

AB': A'B: COS A'BB': tan a;

:

I. Diff. lat. diff. long. :: cos mid. lat. : tan course.

These three proportions comprise the theory of middle latitude sailing, and when to the middle latitude the proper correction, taken from Mr. Workman's table, is added, these theorems will be rendered strictly

accurate.

This is Table XXIX; the middle latitude is to be found in the first column to the left; in a horizontal line with which, and under the given difference of latitude, is inserted the proper correction to be added to the middle latitude to obtain the latitude in which the meridian distance is accurately equal to the departure. The formula for constructing this table is obtained as follows.*

1. A ship, in latitude 51° 18' N., longitude 22° 6' W., has sailed S.

33° 5' E., required her latitude and longitude.

The required latitude is found by plane sailing, as follows:

*The investigation of this formula should be postponed until after reading the

next article, and may be omitted entirely.

In this operation the middle latitude has not been corrected, so that the difference of longitude here determined is not without error. To find the proper correction, look for the given middle latitude, viz., 44° 9' in the table of corrections, the nearest to which we find to be 45°; against this and under 14° diff. of lat. we find 27', also under 15° we find 31', the difference between the two being 4'; hence corresponding to 14° 18′ the correction will be about 28'. Hence the corrected middle latitude is 44° 37', therefore,

cos. corrected mid. lat. 44° 37′ ar. comp. log. 0.14763

therefore, the error in the former result is about 6 miles.

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° 52′ N.