EXAMPLES.

1. A ship from latitude 42° 31′ N., and longitude 58° 51 W., having sailed S.W. by S. 300 miles, required the latitude and longitude at which she has arrived.

To find the diff. of lat. by plane sailing.

2. Required the course and distance from Cape Cod light-house, lat. 42° 3' N., long 70° 4′ W., to the island of St. Mary's, lat. 36° 59′ N., long. 25° 10′ W.

*Enter tab. of mer. parts with the argument 42° 30', and take out the mer. parts corresponding, viz., 2822, which is the meridional diff. of lat. from the equator to 42° 30′ N. Again enter the tab. with the argument 380 21'; the corresponding mer. parts, or mer. diff. of lat., from the equator to lat. 38° 21' N. will be found to be 2495. Subtract the latter from the former, and the remainder will be the meridional diff. of lat. between 38° 21′ N., and 42° 30′ N.

3. A ship from latitude 37° N., and longitude 32° 16′ W., has sailed in a north-westerly direction 300 miles, till she has reached the latitude of 41° N. Required the precise course on which she has sailed, and the longitude at which she has arrived.

Ans. Course 36° 52'; Long. 36° 8'

APPENDIX III.

GREAT CIRCLE SAILING.

1. THE shortest path from one point to another on the surface of a sphere is the arc of a great circle (Geom., App. III., p. 2). A ship, therefore, sailing on the arc of a great circle, joining her point of departure and point of destination on the surface of the earth, will make a shorter voyage than if she sails in the direct course, that is upon the rhumb line joining the same two points.

The practical application of great circle sailing will consist in determining as often as the ship's place is found, that is to say her latitude and longitude, which, under ordinary circumstances, occurs daily, the direction which she ought to take, in order to sail on the great circle from the point where the ship is, to the point of destination. This problem is, in effect, solved in the example on p. 137. In the diagram at that place, s denotes the point where the ship is, s' the point of destination, and the angle rss' the course which the ship ought to steer, in order to sail on the great circle from s to s'. The solution of the problem of Great Circle Sailing, it appears, from the example referred to, requires the application of Napier's Analogies, forms IX. and X. of Art. 86. In a practical treatise on Great Circle Sailing, which appeared in 1846, by S. T. Coit, a table will be found called the "Great Circle Table." It is a table of double entry, in which the logarithm of the ratio of the cosine of the half sum, to the cosine of the half difference, and of the ratio of the sine of the half sum to the sine of the half difference of the colatitudes of s and s', will be found computed for any two latitudes. You enter this table with the less of the given latitudes at top, and the greater at the side; under the former, and on the range of the latter, in the column entitled sine, is found the logarithm of the ratio of the sine of half the sum to the sine of the half difference, and in the column entitled cosine, the ratio of the cosine of the half sum to the cosine of the half difference; if to each of these be added the log. cotangent of the difference of longitude of the point where the ship is, and the point for which she is destined, the results will be the logarithmic tangent of the half sum and half difference of the angles s and s', the former of which, viz., s will be the course upon which the ship should be directed.

The example at p. 137 adapted to this place should read as follows:

1. A ship from lat. 20°, long. 41° 34′ 26′′, is bound to a point in lat. 51° 30', long. 100, upon what course must she sail in order to pursue the shortest path to her destination?

Ans. 30° 28' 12".

2. A ship in lat. 40° 30′ N., long. 700 W., is bound for a place in lat. 51° 22′ N., long. 9° 37′ W., required the course for Great Circle Sailing?

Ans. N. 54° 3' E. The computation of the third side ss' in the same triangle gives the distance sailed.

As a steamer in ordinary weather pursues steadily the course of the great circle from port to port, it may be convenient to calculate beforehand the position of the points in which this great circle intersects the meridians for every five degrees of longitude (five degrees being about the daily progress of a first class steamer), and then if the ship lays upon the direct course for these points successively, it will be sufficient, since the rhumb line differs insensibly from the arc of a great circle for so short a distance. The method of determining these is simple. For when the angle s is calculated, it is only necessary to employ the last two forms, XI. and XII. (Art. 86) of Napier's Analogies, which are used for solving a spherical triangle when two angles and the interjacent side are given.

The data will be the colatitude of the point s, the course pss' previously calculated, and the angle which the meridian PS makes with the meridian whose point of intersection with the great circle course from s to s' is to be calculated; in other words, the difference of longitude of these two meridians.

The logarithm of the ratio of the cosines and sines of the half sum and half difference of the given angles may be taken from " The Circle Table," entering it with the complements of these angles; the log. tangent of the given side will be the same in the calculation for each meridian; the solution of the triangle for each meridian giving the colatitude of the point in which the great circle path intersects the meridian, and the course which the ship ought to take in departing from that meridian.

EXAMPLE.

A ship sailing from the port of New York to Havre or Liverpool, by the shortest path, would steer from Sandy Hook, lat. 40° 27' 30", long. 74° 00′ 48′′ W., E. } S.*running on the rhumb line, and thus give the south shoal of Nantucket a berth of about 15 miles; from this she would sail to a point in lat. 41°, long. 68°, on southern part of George's shoal, in 25 fathoms of water. From this point she would commence Great Circle Sailing, nothing being gained in taking the great circle rather than the rhumb line in the previous short distances. As the great circle from this point to that of destination would pass over Newfoundland, it is recessary to divide the voyage between two great circles, the first terminating at Cape Race, and the second terminating at Cape Clear, the south point of Ireland.

Required the course from the south shoal of Nantucket to George's Shoal, and the points at which the great circle from lat. 41° N., long. 680, W., to Cape Race, in lat. 460, 39' 24", long. 53° 04' 36", intersects the meridians of 600 and 55° W.; and the points in which the great circle from Cape Race to Cape Clear, in lat. 51° 22' N., long. 9° 37′ W., intersects the meridians of 450, 400, 350, 300, 250, 200, and 150 W.

Ans. The course from Nantucket to George's is

This allows for variation of compass.