1919 Mercator Charts - Scale — Page 313, Lesson 283

I want to walk you through how scale works on a Mercator chart. Let's start with a worked example that shows the calculation directly. We have a chart length of 30 centimetres, and that represents an Earth distance of 2078 nautical miles. To find the representative fraction, or RF, we put chart length over Earth distance, both in the same units. So scale equals 1 centimetre per 69.28 nautical miles. But we need centimetres on both sides, so we convert: 1 nautical mile is 1852 metres, and 1 metre is 100 centimetres. That gives us 1 over 69.28 times 1852 times 100, which works out to 1 over 12,831,000. So the scale is 1:12,831,000. Now, the book makes an important point: this calculation has little to do with Mercator charts specifically — it's essentially a scale and departure problem. But there's an alternative method that does use Mercator properties. You can first find the scale at the Equator as a normal RF, then convert that to the scale at 30 degrees South. Let me walk through that alternative. The Earth distance at the Equator is 40 degrees of change in longitude times 60 nautical miles per degree, because the Equator is a great circle. That comes to 2400 nautical miles. Now we put chart length over Earth distance, both in centimetres: 30 centimetres over 2400 times 1852 times 100 centimetres. That gives a scale at the Equator of 1 over 14,816,000. Now we have a normal scale change problem, just like an earlier example. The scale at 30 degrees South equals the scale at the Equator multiplied by the secant of 30 degrees South. Secant is 1 over cosine, so we multiply 1 over 14,816,000 by 1 over cosine of 30 degrees. That gives us 1 over 12,831,000 — the same result as before. Now let's look at Example 5. It seems more complex but is just as straightforward. At 40 degrees North, the scale of a Mercator chart is 1:10,000,000. The question asks: what is the distance in centimetres between the 160 East and 160 West meridians at 20 degrees South? An immediate problem appears: the scale is given at 40 North, but the distance between the two meridians has to be calculated at 20 South. You might be tempted to change the scale from 40 North to 20 South using the scale conversion formula. But that's unnecessary. Think about the Mercator chart — it has parallel meridians, so the distance between two meridians is the same at all latitudes. The problem can be solved completely at 40 North. So you would use the chart length over Earth distance formula, working entirely at 40 North, because the chart distance between those two meridians doesn't change with latitude on a Mercator projection. The scale at 40 North gives you the relationship you need to find that chart distance directly.