We all know that the shortest course on a long sailing passage is a Great Circle route (not precisely correct because the earth is not a perfect sphere, but close enough).
If we stretch a fine thread between two points on a globe of the earth (unless they are True north/south of each other or True east/west on the equator) we can see that the thread crosses each line of longitude or latitude at a different angle. I.e. the course is continually changing. A rhumb line course keeps a constant heading but is not the shortest course. Imagine a thin disk the size of the earth with its center mounted on a ball joint at the earth’s center. By tilting this disk to whatever angle required its edge will trace out a Great Circle course!
For example, from New York (40º 43’ N, 74º 00’ W) to Moscow (55º 45’ N, 37º 37’ E), the Great Circle distance is 4052 Nautical Miles while the rhumb line distance is 4505 NM, a difference of almost 500 miles!

When passage making on a yacht, winds and currents often make us take a course which is not the shortest one, but there are times when we could take a Great Circle route. Over short passages there is little difference between the two and when sailing almost due north or south or east or west near the Equator again very little difference. (A line of longitude is a Great Circle and so is the Equator.)

Strangely, many chart plotters give only rhumb line bearings from our position to a waypoint, not Great Circle bearings and I have not seen a user’s manual, which states whether bearings given are Great Circle or rhumb line routes. How do you know which your plotter or GPS gives?

Very simple as long as you are not on the Equator or very close to it:
When you have your present position, put in a waypoint with exactly your latitude but a longitude a good distance east or west of you, say 100º. Now instruct the GPS to GOTO that waypoint.
If we plotted a course on the standard Mercator chart between our position and the waypoint, it would be either 90º or 270º T. If this is what your GPS or plotter shows, it is giving only a rhumb line bearing.

As an example, my pocket GPS (no names, no pack drill) gave my position in Cape Town as 33º 56.5’ S, 018º 30.9’ E then I instructed it to GOTO a waypoint at 33º 56.5’ S, 118º 00.0’ W. The bearing it gave to the waypoint was 216º T. It was giving me the starting or initial course for Great Circle route! The bearing plotted from a Mercator chart would have been270º T!

If you have a GPS or chart plotter that gives Great Circle bearings, how do you sail a Great Circle route without delving into the navigator’s black art of haversines and cosecants?
The answer is you don’t, but you can sail a very close approximation! Bearing in mind that when sailing almost due north or south or near and almost parallel to the Equator there is little difference between rhumb line and Great Circle, so just follow the GPS.

A Great Circle always starts off curving away from the Equator, but don’t start off exactly on the Great Circle initial bearing or you will start heading off into the wild blue yonder in a few hours. Start your route on a heading about 2º closer to the Equator and after a few hours check the actual route sailed on the chart plotter and trim your course to get onto the course you have chosen. Each day, give the GPS a new GOTO the same waypoint and when there is a change in the bearing, trim your course again to a couple of degrees closer to the Equator. Of course it is wise to go around rocks, reefs and land masses, then do a GOTO again! When you get near the Equator, keep to the GPS bearing because the Great Circle starts flattening out and will curve the other way after crossing the Equator, when you will again start trimming a couple of degrees towards the Equator. When you are about 400 NM from your destination, follow the GPS daily GOTO exactly.

The above will take you on a series of chords, which follow the Great Circle pretty accurately as in the sketch.