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. (Àrt. 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.

41°, long. 68°, on From this point she in taking the great

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. southern part of George's shoal, in 25 fathoms of water. would commence Great Circle Sailing, nothing being gained 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 Lecessary 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, 25, 20, and 150 W.

Ans. The course from Nantucket to George's is

This allows for variation of compass.

The first great circle intersects the meridian of 600 at 44° 24′ 25′′, of 55° at 46° 5' 28" of N. latitude.

The second great circle intersects the meridian of 45° at 48° 55′ 47.5, of 400 at 49° 58' 27", of 35° at 50° 46' 6'', of 30° at 51° 19' 44", of 25° at 51° 40' 3", of 200 at 51° 47′ 28", and of 150 at 51° 42′ 7''.*

The following, from Mr. Coit's work, to which the student is referred for a variety of useful problems, and interesting information, will furnish a number of exercises.

TRACK OF THE ARC OF A GREAT CIRCLE FROM CHESAPEAKE BAY TO BORDEAUX.

Table showing the longitude of the intersections of the latitudes crossed by the arc of a great circle from Cape Henry's light-house, mouth of Chesapeake Bay, to the Corduan light, near Bordeaux; also showing the courses from the points of intersections of the latitudes crossed, and from intersection to intersection.

On this track the difference or saving is 110.

The great circle courses and distances are as follows: From George's Shoal to the meridian of 60°, 57° 14′ and 407 miles. From 600 to 55° N., 62° 40′ E., and 234 miles. From 550 to Cape Race N. 66° 13′ E., and 87 miles. From Cape Race to the meridian of 45 N. 64° 19′ E., and 353 miles. From 45° to 40°, N. 700 18' E., and 205 miles. From 40° to 350, N. 74° 6' E., and 197 miles. From 35° to 30o, N. 77° 57' E., and 191 miles. From 30° to 25o, N. 80° 50' E., and 188 miles. From 250 to 20°, N. 85° 45′ E., and 186 miles. From 200 to 150 N. 89° 42′ E., and 186 miles. From 150 to Cape Clear S. 86° 23', and 202 miles.

For further information on Great Circle Sailing, see "The Practice of Navigation and Nautical Astronomy, by Lieut. Raper, of the Royal Navy," Art. 336, in which work will also be found a convenient table (tab. 5) called the Spherical Traverse table, for solving problems in this kind of sailing.

See also a small collection of "Tables to facilitate Great Circle Sailing," by John Thomas Towson, published by order of the Lords Commissioners of the Admiralty.

SUMNER'S METHOD.

This is a method discovered recently by accident, and consists in calculating the ship's longitude by chronometer for two assumed latitudes, the one of which is the next even degree less, the other the next even degree greater (without odd minutes) than the latitude by dead reckoning. The two positions of the ship thus determined from the longitudes found and the assumed latitudes, being projected on a Mercator's chart, the line joining them passes through the true position of the ship, and any land it may happen to pass through in the vicinity, will have the same bearing from the ship that this line makes with the meridian.

The theory is, that this line is a small portion of what the author terms a parallel of equal altitude, that is, a small circle of the terrestrial sphere, the pole of which is the point of the earth's surface, at which the sun is vertical or in the zenith at the instant of observation. To all places situated on this circle the sun will have the same altitude at the instant. Now since in the two computations in the above problem the latitude only is different, the altitude and declination, which are the other data, remaining the same, the altitude of the sun is therefore the same at the two positions determined, and they are in the same parallel of equal altitude, and as the observed altitude of the ship is also the same, the ship, too, is upon the same parallel of equal altitude, a small are of which may be regarded as a straight line. A perpendicular to the line determined as above will be in the direction of the sun's bearing, and the angle it makes with the meridian will be the sun's azimuth. For the perpendicular to an arc will pass through the pole of the arc.

If two altitudes of the sun be taken, and two lines projected as above, passing each through the place of the ship, its actual position is determined by their intersection.

For the method of allowing for the change of place of the ship between two observations for altitude, and for a variety of problems based upon the principle above explained, see Sumner's work.