42. Advantages of Great-Circle Sailing. In the main there are two advantages in favor of the great-circle track over the rhumb track, viz., the economy in distance and the economy in time.

43. The economy in distance is greatest in high latitudes between places not differing much in latitude, and between places that differ greatly in longitude; it is least between places that are situated nearly on the same meridian, either in north or south latitude, or one in north latitude and the other in south latitude. For instance, the distance on the great-circle track between Cape Agulhas, the southern extremity of Africa, and Perth, Australia, is 4,595 miles, while the distance that has to be traversed by a ship following the rhumb track between the same places is 4,799 miles. A saving of 204 miles is thus effected by sailing along the great circle passing through the two places.

44. The economy in time can only be estimated after having laid out the great-circle track on the chart and considered whether the regions to be passed through, or traversed, along the track

are favorable with regard to winds, currents, etc. The contrast between the greatcircle track and the rhumb track would be more noticeable if it were possible to navigate the Arctic Ocean between two places in the same north latitude, but differing in longitude 180°. Thus, in Fig. 9, let A and B represent two places situated on the same latitude parallel,

FIG. 9

the difference in longitude being 180°, and let P represent the north pole. It is evident, then, that by following the rhumb course a vessel at A had to steer either east or west

in order to reach B; but, by following the great-circle track steer first north from A to P and then

APB, she had to south from P to B.

A glance at the figure will show that the difference in distance, and, consequently, in time, between the two routes would be considerable.

45. Advantage of Great-Circle Sailing to Sailing Ships. The advantages just described apply particularly to steamers that, by virtue of their motive power, can readily adopt the constant changes of courses necessary in greatcircle sailing. But, to sailing ships, great-circle sailing is equally important, which is evident from the following. When a ship, on account of adverse winds, is unable to sail directly to her port of destination, she is put on that tack, or course, that will bring her head nearest the required direction. For instance, if her proper course is N E, the wind NNE, and the ship sails 6 points from the wind when close-hauled, she is put on that tack, or course, that will in the end least lengthen the voyage; in this case the port tack, or 6 points to the right of NNE, or east. The determination of this tack, when about to commence a long voyage-especially in high latitudes-should not be made according to the course obtained by any of the formulas hitherto given, but by the course derived from the calculation, or laying out of the great-circle track.

The term "windward great-circle sailing" is used with special reference to these facts.

46. Towson's Rule.-The rule for windward sailing, as laid down by Towson, a prominent advocate of greatcircle sailing, is as follows: Ascertain the great-circle course and put the ship on that tack which is the nearer to the greatcircle course.

By adhering to this rule the navigator may, at times, find what he considers an adverse wind, according to the rhumb line, to be actually a fair one. The object of great-circle sailing is, therefore, to determine what course or courses must be steered in order that they may coincide with the

great-circle track between the ship and port of destination, and thus render the distance sailed over the least possible.

47. The Great-Circle Track.—If a ship could be kept exactly on the great-circle track when sailing from one port to another, her head would be always pointing toward the port of destination; this would, however, necessitate a continual change of the course, unless the track ran along a meridian or along the equator, which is not very likely.

m

m

Equator

FIG. 10

In order to avoid this constant alteration of courses it is usual, in practice, to use the principle that a small arc of a large circle is practically a straight line; points are therefore taken on the great-circle track at a convenient distance apart and the bearing of each to its next is noted; the ship is then kept on these courses from point to point. Thus, in Fig. 10, if a ship were to sail from A to B along the great-circle track Accd B, her first course would be from A to c, her second from c to c', and so on, until B was reached. In this

manner the advantages to be gained by following the greatcircle track are obtained without frequent change of courses.

48. From the foregoing it is evident that a ship sailing on a great-circle track makes straight for the port of destination and her path through water intersects the meridians Pm, Pn, Po, etc. at an angle that is always varying, whereas by following the rhumb line Arr B, the ship crosses all meridians at the same angle; in other words, the ship's head is kept on the same point of the compass and she never steers directly for the port until it is in sight.

49.

Representation of Track on a

Mercator's

Chart. From this it follows that on a Mercator's chart

the great-circle track must necessarily be represented by a curve, and a little consideration will show that the latter must always lie in a higher latitude than the rhumb track, as shown in Figs. 10 and 20. Thus, if the great-circle track is in the northern hemisphere, it lies nearer to the north pole than the rhumb track; if in the southern hemisphere, it lies

nearer the south pole. This explains how the curved track AVB, Fig. 20, between A and B on a Mercator's chart, represents a shorter distance between the two places than a straight line or rhumb line does; the difference of latitude is the same for both tracks, but the great-circle track has the advantages of the shorter longitude degrees, measured on higher latitude parallels. The higher the latitude the more do the tracks differ, particularly so when the two places are nearly on the same parallel.

50. When the two places are situated in different hemispheres and not on the same meridian, as A and B, Fig. 11, the great-circle track Ancm B between them will be represented on a Mercator chart, as shown in the figure, by a curve of double flexure intersecting the equator E E' at the same point c as the rhumb track between them.

DEFINITIONS RELATING TO GREAT-CIRCLE SAILING

51. The first course sailed on, when following a greatcircle track, is called the initial course and the last one the final course. Thus, in Fig. 10, the angle PAC is the initial course and the angle PBd is the final course.

52. The vertex of a great circle is the point at which the highest latitude is reached, or the point at which the track most nearly approaches the pole and runs perpendicular to the meridian. Thus, in Fig. 10, x is the vertex of the greatcircle track between A and B.

53. It is evident that on every great circle there must be two vertexes and that they must be situated midway between the points of intersection of the great circle and the equator, one in north, and the other in south, latitude. However, it is not necessary to consider more than one vertex, viz., the one relating to the great-circle track. The vertex may or may not be situated between the two places, but, if they are in the same latitude, it lies midway between them.

54. If both of the angles PAC and PBC", Fig. 10, are less than 90°, the vertex will lie between the two places,