1919 Mercator Charts - Scale — Page 313, Lesson 281

Let’s pick this up right where the calculation left off — because the real lesson here isn’t the arithmetic, it’s the logic that lets you skip the arithmetic entirely. There are two learning points I want you to take from that worked answer. The first is practical: on scale questions, the answer options are usually rounded to the nearest convenient number. The exact answer to that scale question was 1:3 258 538, but the closest option offered was 1:3 250 000. That rounding is normal — don’t let it throw you. The second point is the one that saves you time. You don’t actually need to do the calculation at all, because of a fundamental property of the Mercator chart: scale must expand as you move away from the Equator. Think about what that means for the representative fraction. A representative fraction, or RF, is written as 1 over a denominator — like 1:2 000 000. The denominator is the big figure. Now, if scale expands away from the Equator, that means the scale gets larger — the denominator gets smaller. So the denominator must be biggest at the Equator, and it must reduce as you move away from the Equator. In that question, you knew the denominator at 52° South was 2 000 000. Since the Equator is closer to the centre of the Earth than 52° South is, the scale at the Equator must be smaller — which means the denominator at the Equator must be a bigger number. So you’re looking for an option whose denominator is larger than 2 000 000. In that particular set of options, only one answer was smaller in scale — that is, only one had a bigger denominator. That alone would have given you the answer without any calculation. Now let’s formalise this into a proper method. The heading here is: given the Mercator scale at some latitude, find the Mercator scale at some other latitude. There’s a short-cut formula that removes the need to use secants, and removes the need to remember to write the RF as a fraction. You simply need to remember that the large figure in the RF is the denominator — and it’s also the figure we most commonly remember. So a 1:500 000 chart — that’s the ICAO topographical chart — has a denominator, which we’ll call D, of 500 000. The derivations I’m about to show you don’t need to be memorised, but the final formula is important. Here’s how it works. When you compare scales at two different latitudes on a Mercator chart, you can write, for latitude A: scale at A equals scale at the Equator multiplied by the secant of A. And for latitude B: scale at B equals scale at the Equator multiplied by the secant of B. Now divide the two scale formulae. Scale at A over scale at B equals secant A over secant B, because the scale at the Equator cancels out — it’s the same factor in both numerator and denominator. So you get: scale at A over scale at B equals secant A over secant B. And that can be simplified even further — and that simplified form is the formula you’ll actually use. Let me make sure the logic is crystal clear, because this is the heart of it. The Mercator projection is constructed so that scale is correct only at the Equator, and it increases with latitude. The secant of the latitude is the factor by which the scale has expanded at that latitude. So if you know the scale at one latitude, you can find the scale at another by comparing their secants. The Equator itself has a secant of 1 — secant of 0 degrees is 1 — so the scale at the Equator is the base scale, and every other latitude’s scale is that base multiplied by the secant of that latitude. So the practical takeaway: when you’re given the scale at one latitude and asked for the scale at another, you don’t need to compute secants from scratch. You use the ratio of the secants. And when you’re choosing between options, remember the direction of the change — scale expands away from the Equator, so the denominator shrinks as you move away from the Equator, and grows as you move toward it. That directional check alone can often hand you the answer faster than any calculation.