The purpose of this article is to describe a graphical solution to the problems of great-circle sailing. The work is done directly upon a Mercator chart with the aid of a special diagram, part of which is shown in Fig. 1, constructed with the same scale as the chart. This diagram has three sets of lines, one each for tracks, courses, and distances, and may be superimposed over the chart. The portion shown in the figure is sufficient for all tracks in ordinary latitudes when the initial and final positions are both on the same side of the equator. In the examples which follow it is assumed that the base of the diagram is opaque, although speed and accuracy may be increased somewhat by the use of transparent material.
Example 1.—St. Augustine, Florida, to Ferrol, Spain. Rule a straight line on a piece of tracing paper and lay it over the equator of the chart. Mark the points A and B over St. Augustine and Ferrol, as shown in Fig. 2a. Now transfer the tracing to the diagram and slide the ruled line along the equator until the two points lie on the same track, in this case the one whose vertex is in latitude 45° N. The result appears in Fig. 2b. The initial course is then seen to be 55°, and the final course, 103°. There are five complete intervals of ten degrees of distance between A and B, with 5° and 3° remaining at either end.
This gives a total distance of 58°, or 3,480 nautical miles. The track may be traced on the paper and then pricked through onto the chart.
Example 2.—Composite track from Cape of Good Hope eastward, with limiting parallel 50° S. Make a tracing from the chart showing equator and point of departure, A. When this is oriented on the diagram with the point A bisecting the track with vertex at 50°, the course is seen to be 129°, and the distance to the limiting parallel between 42° and 43° (see Fig. 3). The longitude of A is 18°-30' E. By projecting this point onto the equator the longitude of the vertex is found to be 55° farther east, or 73°-30' E.
Example 3.—Howland Island to Cape Flattery. When the point of departure is near the equator the initial course cannot be read in the usual way, but will then differ from the vertex of the track by an odd multiple of 90°. There can be no confusion, however, as inspection of Fig. 4 shows that the tracks radiate out from point A or its vicinity like the points of a true compass rose. Since A and B both lie on the 55° track the course may be read accurately at B by taking the complement, 35°. The distance is 65° and the final course 60°. If the chart is cut between A and B the tracing may be made from any two points with the same latitudes and the same difference of longitude. This is the only case which requires a tracing unless it is desired to have the track drawn on the chart.
Example 4.—Capetown to Sandy Hook. The diagram must now be extended to cover 360° of longitude on both sides of the equator, so that it is four times the size of Fig. 1. On Fig. 5 we read the course from A, 304°, the final course at B, 296°, and the distance, 113°.
Construction of diagram. Any chart projection may be used provided there is a finite or infinite center of symmetry. In the former case the diagram is sometimes known as a spherical co-ordinator. Any convenient parallel will serve in place of the equator for the purpose of orientation. It might be convenient to cut the diagram to fit the chart, and provide some mechanical means of keeping them in register. The lines are plotted by using the co-ordinate scales of the chart and the following formulas:
L is latitude, Lo is longitude, and Lx is the latitude of the vertex of the track. C and D are the course and distance, respectively. The small scale Mercator chart of the world, H.O. 1262b, which measures about 20 by 31 inches, is large enough to be read to the nearest degree by estimation with a diagram showing only 10-degree intervals. For aid in celestial navigation and star identification read local hour angle as difference of longitude; for latitude and declination take latitude of A and latitude of B. Then zenith distance and azimuth are represented by distance and initial course, respectively.
To plan a voyage it will be found convenient to note the points where the ten degree course lines intersect the track. In most cases the mean of the courses at two adjacent points will be very nearly the mercator course between them. Additional points should be added where the distances exceed 20°.