Mercator Charts - Scale — Page 320, Lesson 284

I want to walk you through the scale properties of a Mercator chart, starting with a worked example that shows exactly how we calculate chart length from a real Earth distance, and then moving into the concept of constant scale and its practical limits. Let’s begin with the calculation. We have an Earth distance at 40° North latitude. That Earth distance is called departure. Departure is the distance along a parallel of latitude, measured in nautical miles, between two meridians. The formula for departure is: departure equals change of longitude in minutes multiplied by the cosine of the latitude. In this example, the change of longitude is 2400 minutes, and the latitude is 40° North. So departure equals 2400 times cos 40°, which works out to 1838.4 nautical miles. Now, we want to find the corresponding chart length on a Mercator chart whose stated scale is 1 to 10 million — that is, one unit on the chart represents 10 million of the same units on the Earth’s surface. But there is a catch: the scale on a Mercator chart is only correct at the latitude for which it is defined, and here we are working at 40° North. So we set up the proportion: chart length divided by the Earth distance in nautical miles equals 1 over 10 million. That gives us chart length equals 1838.4 nautical miles divided by 10 million. But that result is in nautical miles, and we need the answer in centimetres. So we convert: one nautical mile is 1852 metres, and one metre is 100 centimetres. Therefore, we multiply 1838.4 by 1852 and then by 100, and divide by 10 million. The answer is 34 centimetres. That is the chart length representing that departure at 40° North on a 1:10 million Mercator chart. Now, let’s move to the concept of constant scale on a Mercator chart. For practical navigation, we consider a chart to have constant scale in any area where the scale is within 1% of the correct scale. That is the professional tolerance: if the scale error is less than or equal to 1%, we treat the scale as constant for practical purposes. On a Mercator chart, scale is constant and correct only along the Equator. That is the one line where the chart scale exactly matches the stated scale. Everywhere else, the scale expands as latitude increases because the chart stretches the parallels to keep rhumb lines straight. However, there is a useful band where the scale is within that 1% tolerance. Specifically, the scale is within 1% of correct scale within a band of 8° of latitude either side of the Equator. So from 8° South to 8° North, the scale error is less than 1%, and we can treat the chart as having effectively constant scale for practical navigation. There is a note for exam purposes: the 8° figure can be quoted as 480 nautical miles out of 500 nautical miles. That is a ratio — 480/500 — which represents the same 1% tolerance in a different form. So if you see 480/500 NM in an exam context, it refers to this 8° band either side of the Equator. Let me summarise the key points. Departure is change of longitude in minutes times cosine latitude. Chart length on a Mercator chart is calculated by converting the Earth distance to the same units as the chart scale. Scale is constant and correct only at the Equator. For practical navigation, scale is considered constant within 1% of correct scale, which occurs within 8° of latitude either side of the Equator, also expressed as 480/500 NM.