Let's pick this up right where the previous chapter left off — Page 313, Lesson 277

Let's pick this up right where the previous chapter left off. We've just established the general principles of Mercator scale, and now I want to formalise that into the working equation you'll actually use. The key starting point is the departure equation. On a cylindrical projection, the meridians must be drawn as parallel straight lines. That means the east-west distance between any two meridians on the chart is a fixed value, regardless of latitude. But on the Earth, the actual distance between those meridians shrinks as you move away from the Equator. So the chart's east-west scale at any latitude is defined by the departure equation — the scale has to expand to keep those meridians parallel. By rearranging that equation, we find that the east-west scale depends on the secant of the latitude. Secant is the reciprocal of cosine — so at the Equator, where latitude is zero, cosine is 1 and the secant is 1, giving no expansion. As latitude increases, cosine decreases, so the secant grows, and the scale expands. Now, Mercator's clever step: he adjusted the north-south spacing of the parallels of latitude so that it also becomes a function of the secant of the latitude. That way, the north-south and east-west scale expansions happen at exactly the same rate. When both directions expand identically, the projection becomes orthomorphic — meaning shapes are preserved — though it's non-perspective, because it's not a simple geometric projection from a point. So the core principle to hold onto: on a Mercator chart, scale expands away from the Equator, and that expansion is proportional to the secant of the latitude. That gives us the equation we finish up with. Let me write it out for you: Scale at latitude φ = Scale at Equator × sec φ Where φ is the latitude. So if you know the scale at the Equator, you multiply by the secant of the latitude to get the scale there. And if you know the scale at some latitude, you divide by the secant to get back to the Equator. That single relationship drives every calculation in this chapter — finding scale at another latitude, finding the Equator scale, even working out the area of "constant" scale. We'll work through each of those cases next.