Here’s the setup — Page 451, Lesson 434

Let’s look at what’s actually on this page, because it’s not prose — it’s a table of numbers, and it’s one of the most important tables in general navigation. What you’re seeing is a convergency table, and it’s used to convert between great-circle and rhumb-line directions. Here’s the setup. On a chart, when you draw a straight line between two points, that line is a rhumb line — it crosses every meridian at the same angle. But the shortest path between two points on the Earth’s surface is a great circle, and a great circle crosses meridians at changing angles. The difference between those two directions at any point is called convergency, and this table gives you that correction. The table is organised by latitude down the left-hand side — you can see the rows labelled N 60, N 50, N 20, N 10, 0, S 10, S 30, S 52, S 60, and so on. So the top rows are northern latitudes, the middle is the equator at 0, and the lower rows are southern latitudes. The columns across the top are longitude differences — the angular distance in degrees between your two meridians. Here’s how you read it. You find your mean latitude — the average of the two latitudes you’re working between — and you find the difference in longitude between your two points. Where that row and column meet, the number in the table is the convergency in degrees and minutes. Let me walk you through a specific example so you can see the pattern. Look at the row for N 60. The values start around 23 degrees and climb as you move across the columns — 23 24, 22 57, 22 37, 22 20, 22 04, 21 49, 21 35 — and they keep decreasing as the longitude difference gets smaller. That’s the key relationship: the greater the difference in longitude, the greater the convergency. And the higher the latitude, the greater the convergency for the same longitude difference. At the equator, the values drop to around 30 degrees and below — you can see the row at 0 latitude shows values like 29, 30, 31, 32 — because at the equator, meridians are parallel and convergency is at its minimum. Now, the numbers themselves are in degrees and minutes. So a value like 23 24 means 23 degrees and 24 minutes of convergency. A value like 21 35 means 21 degrees and 35 minutes. When you see a single number like 30 or 32 in the southern rows, that’s the convergency in whole degrees. Here’s the practical use. When you’re navigating, you often measure a direction on the chart as a rhumb-line bearing, but for long-distance great-circle navigation you need the great-circle direction. The convergency from this table is the correction you apply. The rule is: convergency equals the change in great-circle direction minus the change in rhumb-line direction. In other words, the great-circle track changes direction by the convergency amount more than the rhumb line does, over the same longitude difference. So the table gives you that correction in degrees and minutes, and you apply it with the correct sign — add it when you’re converting from rhumb line to great circle in one direction, subtract it when you’re going the other way, depending on whether you’re in northern or southern latitudes and whether you’re travelling east or west. Let me give you a concrete read from the table so you can see the mechanics. Look at the row for N 50. The first value is 20 57 — that’s 20 degrees 57 minutes of convergency for a large longitude difference. As you move right, the values drop: 20 57, 20 55, 20 53, 20 51, 20 48, 20 45, 20 42, 20 38, 20 33, 20 29, 20 24, 20 18, 20 13, 20 07, 20 01, 19 55. So at N 50, for a smaller longitude difference, convergency drops to around 19 degrees 55 minutes. Now look at the southern rows. At S 60, the values are much smaller — 15 58, 16 01, 16 03, 16 06, 16 10, 16 14, 16 18, 16 23, 16 28, 16 33, 16 38, 16 44, 16 50, 16 56, 17 02, 17 08, 17 14. So at S 60, convergency ranges from about 15 degrees 58 minutes up to 17 degrees 14 minutes, depending on the longitude difference. Here’s the critical thing to remember about this table: it’s symmetrical about the equator in terms of the pattern, but the values are not identical north and south. The convergency at a given latitude and longitude difference is the same magnitude whether you’re north or south of the equator — the table just lists them separately so you can read directly without interpolating across the equator. One more thing to notice. The table is not linear — the values don’t decrease by a constant amount as you move across the columns. Look at the N 60 row again: 23 24, 22 57, 22 37, 22 20, 22 04, 21 49, 21 35. The differences between successive values are 27 minutes, then 20, then 17, then 16, then 15, then 14. So the convergency changes more rapidly at large longitude differences and more slowly at small ones. That’s because convergency is a sine function of latitude — it’s proportional to the sine of the mean latitude times the difference in longitude. So when you use this table in flight planning, you’re doing this: you take your two points, you find the mean latitude, you find the difference in longitude, you read the convergency from the table, and you apply it as the correction between your great-circle and rhumb-line directions. That correction is what lets you plot a great-circle track on a chart that only shows rhumb lines as straight lines. That’s the whole function of this page. It’s a reference table, but understanding how it’s organised — latitude down the side, longitude difference across the top, convergency in degrees and minutes in the cells — is what makes it usable in real navigation.