Scale at any latitude (λ) = Scale at Equator × sec (λ) — Page 313, Lesson 279

We're starting a new topic now: Mercator scale problems. This is the heart of how we work with Mercator charts, and it all hangs on one formula. Let me give it to you exactly as it appears in the book: Scale at any latitude (λ) = Scale at Equator × sec (λ). Let me unpack that. λ, the Greek letter lambda, is the symbol for latitude. Sec is the secant, which is the reciprocal of cosine — so sec(λ) = 1 / cos(λ). What this formula tells us is that on a Mercator chart, the scale isn't constant. It expands as you move away from the Equator toward the poles. At the Equator itself, the scale is at its base value. Everywhere else, you multiply that base scale by the secant of the latitude. Now, the book tells me that all Mercator scale questions come in one of four basic types. Let me list them so you recognize them when you see them. First: given the scale at the Equator, find the Mercator scale at some other latitude. Second: given the Mercator scale at some latitude, find the scale at the Equator. Third: given the Mercator scale at some latitude, find the Mercator scale at some other latitude. And fourth: given a fixed chart distance between meridians, find the Mercator scale at a specific latitude. Let's work through the first type with Example 1. The question: if the scale of a Mercator chart at the Equator is 1:1,000,000, what is the scale at 60N or 60S? The options are 1:2,000,000, 1:1,000,000, 1:866,000, or 1:500,000. Here's a critical rule the book stresses: whenever you solve problems using representative fractions — that's what RF stands for — the scale must be written mathematically in fraction form. So 1:1,000,000 becomes the fraction 1/1,000,000. Don't skip that step; it's what makes the math work. So we set up: Scale at 60N = Scale at Equator × secant 60N. That gives us 1/1,000,000 × secant 60°. But the book points out it's easier to write secant as 1/cos. Why? Because that way we keep 1 as the numerator, which is what we want for a representative fraction, and we only have to multiply out the denominators. So secant 60° becomes 1/cos 60°, and cos 60° is 0.5. So we have 1/1,000,000 × 1/0.5. Multiply the denominators: 1,000,000 × 0.5 = 500,000. So the scale at 60N is 1/500,000, which is 1:500,000. That's option (d). Now, the book flags a trap here, and I want you to really absorb this. At 60°N or S, the scale on a Mercator chart is double the scale at the Equator. If the scale doubles, the denominator of the scale is halved. Remember this relationship: if scale expands, the denominator decreases; if scale contracts, the denominator increases. That's counterintuitive at first — a bigger scale means a smaller number in the denominator. So 1:1,000,000 at the Equator becomes 1:500,000 at 60°, because the scale has doubled. Now let's move to the second type of problem: given the Mercator scale at some latitude, find the scale at the Equator. Example 2: if the scale of a Mercator chart at 52S is 1:2,000,000, what is the scale at the Equator? Options are 1:3,250,000, 1:1,000,000, 1:866,000, or 1:500,000. We start the usual way: Scale at 52S = Scale at Equator × secant 52S. Now substitute what we know. We know the scale at 52S is 1/2,000,000. So we write 1/2,000,000 = Scale at Equator × 1/cosine 52°. Now we re-arrange to solve for the scale at the Equator. Multiply both sides by cosine 52°, and we get: Scale at Equator = cosine 52° / 2,000,000. Now here's a practical calculator tip from the book. It's easier to enter this into your calculator denominator first, then divide by the numerator. That gives you the value of the denominator with 1 as the numerator, which is what we want for a representative fraction. So key in: 2,000,000 ÷ cos 52. That gives you 3,248,538. You write it down as 1/3,248,538, which is 1:3,248,538. The answer is therefore option (a), 1:3,250,000 — they've rounded it. So you see the pattern: the formula is your anchor, and the two skills are writing secant as 1/cos, and remembering that when you solve for the Equator scale, you're dividing by the cosine. Let's keep going when you're ready.