= 16° ch.long × sin 42.5° = 10.8° — Page 247, Lesson 228

Let’s pick this up right where the calculation left off. We’ve just worked out that the convergency between the two meridians is 10.8 degrees — that came from 16 degrees of longitude change multiplied by the sine of 42.5 degrees latitude. Now, the key idea I want you to hold onto is this: convergency can be thought of in two completely equivalent ways, and the book gives you both. The first way is to say that convergency is the change in great circle track angle. So if we’re tracking along a great circle from H to G, and we know the track angle at H, then the track angle at G is that initial angle plus the convergency. In our example, the track angle at G going westwards is 260.8 degrees true. The reciprocal — that is, the track back from G to H — is therefore 080.8 degrees true. That’s just 260.8 minus 180. The second way to think about convergency is as the angle of inclination of the meridians. This is the geometric picture. Imagine drawing a parallel construction line — a line that stays parallel to the meridian at H, running across to G. That construction line is not a great circle; it’s just a reference. From G to H along that parallel construction line, the reciprocal track is 070 degrees. Now, the convergency between that construction line and the actual meridian at G is our 10.8 degrees. So to get the true track from G back to H, measured from True North — which is the meridian at G — we add those together: 070 plus 10.8 gives us 080.8 degrees true. So you see, both methods land on the same answer, 080.8 degrees true, because they’re just two ways of describing the same geometric fact: the meridians converge toward the pole, and that convergence is exactly the convergency we calculated. That figure shows a great circle cutting two meridians, X and Y, with tangents drawn to those meridians — that’s the visual proof of why convergency equals the change in track angle. The tangent at each meridian points along True North at that location, and the angle between those tangents is the convergency. So the takeaway for your navigation work: whenever you’re converting a great circle track from one meridian to another, you add or subtract the convergency, and you can always check it by drawing that parallel construction line and adding the inclination angle. Both give you the same true track.