36 miles; 6th, S. E. by E. 58 miles: required the direct course, and distance made good.

Direct course S. 25° 59′ E., or S. S. E. E., nearly. Ans. {Distance 95.87 miles.

6. A ship sails, 1st, N. W. by W. W. 40 miles; 2d N. W. by N., 41 miles; 3d, N. by E. 16.1 miles; and 4th, N. E. E. 32.5 miles: required the distance made, and the direct course.

Ans. Course, 21° 54′ West of North. Dist. 94.6 miles. These examples will, perhaps, suffice to illustrate the principles of plane sailing.

The longitude, made on any course, cannot be determined by these methods, for this being the arc of the equator intercepted between two meridians, cannot be found under the supposition that the meridians are parallel.

The most simple case of finding the difference of longitude is when the ship sails due east or due west: this is called Parallel Sailing.

SECTION IV.

PARALLEL SAILING.

17. The entire theory of parallel sailing is comprehended in the following proposition, viz.:

The cosine of the latitude of the parallel, is to radius, as the distance run to the difference of longitude..

Let IQH represent the equator, and FDN any parallel of latitude: then, CI will be the radius of the equator, and EF the radius of the parallel.

Suppose FD to be the distance sailed, then the difference of longitude will be measured by IQ, the arc intercept

H

N

E

D

18. If we denote by D the distance between any two meridians, measured on the parallel whose latitude is L; and by D the distance between the same meridians measured on the parallel whose latitude is L', the arcs are similar, and we shall have (Geom., Bk. V., Prop. 14), cos L : D :: COS L' : D',

Hence, when the longitude made on different parallels is the same, the distances sailed are proportional to the cosines of the parallels of latitude.

19. By referring to Th. V., Bk. I., we see that in any right-angled triangle

R cos angle at base ::

or COS E R ::

:

hyp. base, G EC : EG;

and by comparing this with the propor

tion,

:

dist.

diff. long.

lat.

cos lat. : R :: dist. diff. long; we see, that if in a right-angled triangle the angle at the base be made equal to the latitude of the parallel, and the base to the distance run; then, the hypothenuse will represent the difference of longitude.

C

B

It follows therefore, that any problem in parallel sailing, may be solved as a simple case of plane sailing. For, if we regard the latitude as the course, the distance run as the base, the difference of longitude will be the hypo

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 sails 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 Art. 18,

As cos lat. 71° 10' arith. comp. 0.491044

3. A ship in latitude 32° N. sails due E. till her dif ference of longitude is 384 miles: required the distance Ans. 325.6 miles.

run.

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

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 rectified.

may be

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 O 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.

Τ

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 TT, 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.