Mercator Charts - Properties — Page 297, Lesson 264

Let’s start with what this projection actually is. The Mercator projection we’re dealing with here is called a normal or direct Mercator. The projection surface—the cylinder—touches the Reduced Earth at the Equator. And here’s a key limitation: the geographic poles cannot be projected, because they lie on the axis of the cylinder. So you can never show the poles on a normal Mercator chart. Now, the heart of this section is scale. The fundamental rule is: Mercator scale expands as the secant of the latitude. Let me unpack that, because it’s the single most important idea here. It all comes from the departure formula. You’ll remember that: Departure = change of longitude (in minutes) × cos latitude Departure is the east–west distance on the Earth’s surface. Now, Mercator’s insight was this: because the meridians are drawn as parallel straight lines on a cylindrical projection, a given change of longitude is always represented by the same distance on the chart—so many centimetres—no matter what the latitude. So the chart length for a change of longitude is held constant. What varies is the Earth distance, the departure, because that depends on latitude. So we rearrange the formula to see how the scale changes: change of longitude = Departure ÷ cos latitude Now, in mathematics there’s a trigonometrical function called the secant. Its definition is: secant θ = 1 ÷ cosine θ So we can rewrite the equation as: change of longitude = Departure × secant latitude Here’s the reasoning chain: because the meridians are parallel at all latitudes, the east–west scale has to change as a function of the secant of the latitude. And Mercator realized that the north–south scale has to change by exactly the same amount. That gives us the key equation: scale at any latitude (λ) = scale at Equator × sec(λ) Let’s work a practical example. Suppose the scale at the Equator is 1/1,000,000. What’s the scale at 8°N—or, by symmetry, 8°S? scale at 8°N = scale at Equator × sec 8°N Now, there’s a useful trick here. There’s no point evaluating secant as a numerator. When we multiply representative fractions, we want a ‘1’ on the top line. So we’re better off writing secant as 1/cos, because then we’ll be multiplying denominators. So: scale at 8°N = (1/1,000,000) × (1/0.990268) = 1/990,268 That means the scale at 8°N—or 8°S—is 99% of the scale at the Equator, or within 1% of correct scale. And that 1% figure is genuinely important in navigation. Here’s the practical rule: up to a scale error of 1%, we can treat a chart as having constant scale, which means you can measure distances directly with a ruler. But once the scale error exceeds 1%, you must find distances either by calculation, or by using the local latitude scale and measuring small distances at a time with a pair of dividers. So the figure to remember—and I want you to commit this to memory—is: Mercator scale is within 1% up to 8° from the Equator. That’s the boundary that separates “measure with a ruler” from “measure with dividers or calculate.”