Let me set the scene — Page 451, Lesson 438

I want to walk you through what this page is showing, because at first glance it looks like a wall of numbers, but it's actually one of the most important charts you'll use in general navigation — the convergency diagram for the Earth's meridians. Let me set the scene. This is a table that gives you the convergency — the angle at which two meridians meet at the pole — for different latitudes and for different differences of longitude. Convergency is the amount by which meridians, which are parallel at the equator, come together and eventually meet at the poles. The whole point of this chart is to let you look up that value quickly without doing spherical trigonometry every time. Now, let's read the structure. Down the left-hand side, you have latitude, running from N 50 at the top, through N 20, N 10, 0 (the equator), S 10, S 30, S 52, and S 60 at the bottom. So the rows are latitudes, both north and south of the equator. Across the top, you have difference of longitude — the values run from 25 degrees down through 18, 11, 04, and so on, all the way down to 17 degrees at the far right. So the columns are differences of longitude. Here's the key idea: you enter the table with your latitude and your difference of longitude, and the number at the intersection is the convergency in degrees. Let me give you a concrete example so you see how it works. Look at the row for N 50. The first column is a difference of longitude of 25 degrees. The convergency there is 19 19 19 14 19 08 19 02 18 56 — wait, let me be careful. Actually, the numbers in the body are the convergency values, and they're given in degrees and minutes. So at N 50 with a difference of longitude of 25 degrees, the convergency is 19 degrees 19 minutes. As the difference of longitude decreases across the row — 19, 14, 08, 02, then 18 56, 18 49, 18 43, and so on — the convergency decreases too. That makes sense: the smaller the difference of longitude, the smaller the angle between the meridians. Now here's a really important pattern I want you to notice. Look at the equator row, marked 0. The convergency values there are tiny — 18 08, 18 07, 18 06, 18 06, 18 05, 18 04 — and they barely change as you move across. That's because at the equator, meridians are essentially parallel, so convergency is at its minimum. The values are small and nearly constant. Now look at the S 60 row at the bottom. The convergency values are much larger — 16 29, 16 36, 16 43, 16 50, 16 57, 17 04, 17 11, 17 18, 17 25, 17 32, 17 39, 17 46, 17 53, 18 00, 18 07, 18 14, 18 21. At 60 degrees south, the meridians are converging rapidly toward the pole, so the convergency is much bigger for the same difference of longitude. This is the fundamental relationship: convergency increases as latitude increases, and convergency increases as difference of longitude increases. Let me also point out the intermediate rows so you see the progression. At S 30, the values run from 17 35 up through 17 37, 17 39, 17 40, 17 42, 17 44, 17 45, 17 47, 17 48, 17 50, 17 52, 17 53, 17 55, 17 56, 17 58, 18 00, 18 02. At S 52, they run from 16 56 up through 17 00, 17 05, 17 10, 17 15, 17 20, 17 24, 17 29, 17 34, 17 39, 17 44, 17 49, 17 53, 17 58, 18 03, 18 08, 18 13. And at S 60, as I said, they're the largest of all. So the way you use this in practice: you find your latitude on the left, you find your difference of longitude along the top, and you read the convergency at the intersection. That convergency value is what you'll use when you're converting between great-circle and rhumb-line tracks, or when you're working out the difference between initial and final great-circle courses. It's a lookup table that saves you from doing the spherical trig every single time. One more thing to notice — the values are given in degrees and minutes, like 19 19 meaning 19 degrees 19 minutes, and 18 56 meaning 18 degrees 56 minutes. So when you read a value, you're reading two parts: the degrees and the minutes of arc. That's the whole chart. It's a convergency table: latitude down the side, difference of longitude across the top, and the convergency in degrees and minutes at the intersection, with the values growing as you move toward the poles and as the difference of longitude grows.