5. Two ships in the parallel of 47° 54′ N., have 9° 35′ difference of longitude, and they both sail directly S., a distance of 836 miles: required their distance from each other at the parallel left, and' at that reached.

Ans. 385.5 miles, and 479.9 miles.

SECTION V.

MIDDLE LATITUDE SAILING.

20. 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 which 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.

passing through O and T, measured on the parallel of lati

The middle latitude is half the sum of the two extreme latitudes, if they are both of the same name, and half their difference, if they are of contrary names.

The supposition above 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 OT cuts the successive meridians at a very small angle.

The principal approaches nearer to accuracy as the angle 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 parallel M'M. But still, in very high latitudes, a small advance in latitude makes a considerable difference in meridional distance; hence, this principle is not to be used 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 results of it rendered in all cases accurate. This table we have given at the end of this work.

21. The rules for middle latitude sailing may be thus deduced.

T

dif long.

dep.

T

We have seen, in the first case of plane sailing, that if a ship sails on an oblique rhumb from 0 to T, that the hypothenuse OT will represent the distance; OT" the difference of latitude, and T'T, the departure. Now, by the present hypothesis, the departure T'T is equal to the middle parallel of latitude between the meridians of the places sailed from and arrived at: so that the difference of longitude of these two places is the same as if the ship had sailed the distance TT on the middle parallel of latitude. The determination of the difference of longitude is, therefore, reduced to the case of parallel sailing: for, T'T now representing the distance on the

dist.

We

that parallel, we shall have, by the last case, the difference of longitude represented by the hypothenuse OʻT. therefore have the following theorems:

diff. longitude.

TT';

that is, since sin O' = cos 0TT'

cos mid. lat. : distance :: sin. course

III. In the triangle OTT", we have

R tangent 0 ::
:

OT' :

comparing this with the first proportion, and observing that the extremes of this are the means of that, we have OT : ᎤᎢ :: cos ᎤᎢᎢ : tang 0;

that is, diff. lat. : diff. long. :: cos mid. lat. : tang course.

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

In the table of pages 93 and 94, the middle latitude is to be found in the first column to the left. Then, along the horizontal line, 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. Thus, if the middle latitude be 37°, and the difference of latitude 18°, the correction will be found on page 94, and is equal to 0° 40'.

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 dis

tance, as also the difference of longitude between the two

places ?

Lat. from 51° 18' N.
Lat. to 37°

0 N.

18 = 858 miles.

6.989700 | Cos mid lat 44° 9' ar c 0.144167 10.000000 tang course 33° 5' 9.813899 2.933487:: diff. lat. 858

2.933487

9.923187 diff. long. 779 2.891553

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'; and 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. 0.147629

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 lat itude come to.

Ans.

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., aving 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?

Ans. Course N. 33° 19′ W., or N. W. nearly;

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′ W.; the middle latitude being corrected by Workman's table.

Ans. Course N. 51° 11′ E.; dist. 1189 miles.

MERCATOR'S SAILING.

22. It has already been observed, that when a ship sails on an oblique rhumb, the departure, the difference of latitude, and the distance run, are truly represented by the sides of a right-angled triangle.

Thus, if a ship sails from A to B, the departure B'B will represent the sum of all the very small meridian distances, or elementary departures, b'b, p'p, &c.; the difference of latitude AB' will represent, in like manner, the small differences of latitude Ab', b'p', &c.; and the hypothenuse AB, will express the sum of the distances corresponding to these several differences of latitude

B'

B

P

b

and departure. Each of these elements is supposed to be taken so small, as to form on the surface of the sphere a series of triangles, differing insensibly from plane triangles.

Let ABB' be a triangle, in which the angle A represents the course, AB' the difference of latitude, B'B the departure, and AB the distance run. Produce the side AB' to C', until CC' shall be equal to the difference of longitude of the two extremities of the course: then, for the sake of distinction, we call

AB' the proper difference of latitude,

=

AC the meridional difference of latitude,

=

and we are now to explain the manner of constructing a table, called a table of meridional parts, which will furnish the meridional differences of latitude when the proper differ ences are known.

Let Ab'b represent one of the elementary triangles; b'b will then be one of the elements of departure; and Ab' the corresponding difference of latitude. Now, as b'b is a