In shorthand: angle of inclination = ch.long × sine of parallel of origin — Page 340, Lesson 303

Let’s start with the key formula, because everything in this lesson hangs off it. The angle of inclination of the meridians equals change of longitude times the sine of the parallel of origin. In shorthand: angle of inclination = ch.long × sine of parallel of origin. Now, that angle of inclination of the meridians has a proper name in navigation — it’s called chart convergence. So we can rewrite the formula as: Chart convergence = change of longitude × sine of parallel of origin. Let me unpack what each piece means. Change of longitude, ch.long, is simply how many degrees of longitude you’ve moved across the Earth — the difference in longitude between two places. The parallel of origin is a specific latitude line on the chart — we’ll come back to what makes it special. And sine is just the trigonometric sine function of that latitude angle. So the formula says: the amount the meridians converge on the chart is proportional to how far east–west you’ve travelled, scaled by the sine of that origin latitude. Let me give you the concrete numbers from the figures. In Figure 21.4, a change of longitude of 100° is represented on the chart by an angle of inclination of the meridians of 100° × sin 45°, which works out to 70.71°. In Figure 21.5, a change of longitude of just 10° gives 10° × sin 45°, which is 7.071°. Notice the relationship is linear — ten times the longitude change gives ten times the convergence. Now, that sine of the parallel of origin has its own name. It’s called “the constant of the cone”, and it’s given the symbol n. So whenever you see n in this projection work, it’s that sine value — the constant of the cone. Let’s move to the Simple Conic Projection and its scale behaviour. On the simple conic, the scale is correct at the parallel of tangency — that’s the latitude where the paper cone actually touches the Reduced Earth. At that one line, distances on the chart match distances on the Earth. But here’s the problem: the scale expands away from the parallel of tangency rather rapidly. So the further you get from that line, the more the chart stretches distances. That’s a serious distortion for navigation. That’s exactly what Lambert set out to fix. He modified the simple conic to reduce this rapid rate of scale change. Instead of having the cone touch the Reduced Earth at one line, Lambert made the cone of his projection go inside the Reduced Earth. The cone now cuts through the Earth rather than just touching it. The result: the scale is now correct at two points — two lines of latitude where the cone intersects the Earth. These two lines are called the Standard Parallels. So now we have a scale story in three zones. Outside the standard parallels, the scale expands — it’s greater, more expanded than on the standard parallels. Between the two standard parallels, the scale contracts — it’s less, contracted. And the scale is least of all on the parallel of origin. Wait — what happened to the old parallel of tangency? It’s been renamed. The old parallel of tangency of the simple conic is now called the parallel of origin of the new Lambert projection. And just as on the simple conic, the scale is least on this parallel. The whole point of this redesign is to even out the scale error. By having scale expand in one region and contract in another, Lambert made the chart much more of a constant scale chart — the scale variation across the sheet is far smaller than on the simple conic. Now, there’s a practical rule for laying out these standard parallels, called the one sixth rule. In Figure 21.12, the standard parallels are 43°N and 47°N, and the parallel of origin is 45°N — right in the middle. The upper standard parallel appears one sixth of the way from the top of the chart, and the lower standard parallel appears one sixth of the way from the bottom. That positioning ensures minimum scale variation over the sheet. So let me pull it all together. The simple conic had one line of correct scale — the parallel of tangency — and scale expanded rapidly away from it. Lambert pushed the cone inside the Earth, giving two standard parallels of correct scale, with expansion outside them and contraction between them, least of all on the parallel of origin. And the convergence of meridians on the chart is governed by that formula: chart convergence equals change of longitude times the sine of the parallel of origin, where that sine is the constant of the cone, n.