But cB is the cosine of the latitude IB to the radius CI, that is, cB is CI times the trigonometrical cosine of the latitude; hence the above proportion is

CIX cos. lat. CI :: distance: diff. long.

... cos. lat. : Rad. (=1): : distance: diff. long.

....

(1). Corollary: hence if the distance between any two meridians, measured n a parallel in latitude L be D, and the distance of the same meridians, measured on a parallel, in latitude L' be D', we shall have, (Geom., prop. 15, Cor. 2, Book 5,)

diff long

Mat.

Hence if one of the legs of a right-angled triangle represent the distance run on any parallel, and the adjacent acuteangle be equal to the degrees of lat. of that parallel, then the hypotenuse will represent the difference of longitude, since this hypotenuse will be determined by the foregoing dist. proportion (1). It follows, therefore, that any problem in parallel sailing may be solved by the traverse table, computed to degrees, as a simple case of plane sailing; for by considering the latitude as the course, and the distance as the difference of latitude, the corresponding distance in the table will express 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.

2. If a ship sail E. 126 miles, from the North Cape, in lat. 71° 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' S.

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.

385.5 miles, and 479.9 miles.

Middle Latitude Sailing.

(67.) 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.

Middle latitude sailing proceeds on the supposition that the departure or sum of all the meridional distances

b'b, c'c, d'd, &c. from A to B, is equal to the distance M'M of the meridians of A and B, measured on the middle parallel of latitude between A and B.

M

This supposition becomes very inaccurate when the course is small, and the distance run great; for it is plain that the middle latitude distance will receive a much greater accession than the departure, if the track of 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 still, as 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, recently constructed by Mr. Workman, and judiciously introduced by Mr. Riddle in the second edition of his valuable Treatise on Navigation, the imperfections of the middle latitude method may be removed, and the results of it rendered in all cases accurate. This table we have given at the end of the present volume, and have explained its construction in the introductory explanation to the Tables.

The rules for middle latitude sailing may be thus deduced.

A

B

It has been seen at (64) that the difference of latitude, departure, and distance, sailed on any oblique rhumb, will be all accurately represented by the sides AB', B′B, AB, of a 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'B, on the middle latitude parallel; the determination of this difference of longitude is, therefore, reduced to a case of parallel sailing, for BB', now representing the distance on the parallel, and an angle A' BB' being made equal to the latitude of that parallel, we shall have the difference of longitude, represented by the hypotenuse A'B. We thus have the following theorems, viz. in the triangle A'B'B,

that is,

cos. A'BB': BB' :: radius : BA';

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

In the triangle A'BA,

that is,

sin. A': AB:: sin. A: A'B;

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

111. 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 ac

curate.

EXAMPLES.

1. A ship, in latitude 51° 18′ N., longitude 22° 6′ W., is bound to a place in the S.E. quarter, 1024 miles distant, and in lat. 37° N.: what is her direct course and distance, as also the difference of longitude between the two places;

To

In this operation the middle latitude has not been corrected, so that the difference of longitude here determined is not without error. 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. 0·1483780

therefore, the error in the former result is about 7 miles.