Let me start with the general principles — Page 313, Lesson 277

I want to walk you through the scale principles of the Mercator chart. This is a new chapter, Chapter 19, and it builds directly on what we covered in the previous chapter about cylindrical projections. Let me start with the general principles. On a Mercator chart, the meridians are drawn as parallel straight lines. That's a fundamental feature of any cylindrical projection. Because the meridians are parallel, the east‑west scale at any given latitude is governed by what we call the departure equation. Departure is the distance along a parallel of latitude between two meridians, and on the chart the distance between those same two meridians is fixed — it doesn't change with latitude. So if you rearrange that equation, you find that the east‑west scale is proportional to the secant of the latitude. Secant is simply 1 divided by the cosine of the angle. Now, Mercator faced a problem: if only the east‑west scale changed with latitude, shapes would be distorted. So he devised a solution. He adjusted the north‑south spacing of the parallels of latitude so that it also varies as a function of the secant of the latitude. By making both the north‑south and the east‑west scale expand at the same rate, the projection became orthomorphic. Orthomorphic means that angles are preserved locally — a small shape on the chart is similar to the corresponding shape on the Earth. But note: this is a non‑perspective projection. You cannot imagine it as a simple geometric projection from a light source; it's a mathematical construction. The key takeaway is this: on a Mercator chart, scale expands as you move away from the Equator. The expansion is proportional to the secant of the latitude. So at the Equator, where the latitude is zero degrees, the secant of zero is 1, and the scale is at its smallest. As you go north or south, the secant increases, and so does the scale. That brings us to the equation that summarises this relationship. The equation is the core tool we'll use for all the calculations in this chapter — finding scale at one latitude given scale at another, or given scale at the Equator, and so on. We'll work through each of those cases in the sections that follow.