DenominatorA — Page 313, Lesson 282

Let’s pick this up right where the formula lives, because that equation is the whole engine of this page. I want you to look at the equation we’re told to remember: Denominator A over Denominator B equals cosine A over cosine B. So, written out: DenominatorA / DenominatorB = cos A / cos B. Now, you may recall this exact shape from earlier in your navigation work — it’s the formula used to compare the departure between two longitudes at different latitudes. Departure, remember, is the east–west distance along a parallel of latitude. The only difference now is that D, the letter we’re using, represents the denominator of scale rather than the departure. So instead of comparing distances, we’re comparing scale denominators. Let me make sure the meaning of “denominator of scale” is crystal clear. A scale of 1:2,000,000 means one unit on the chart equals two million of the same units on the Earth. The denominator is that big number — 2,000,000. A larger denominator means a smaller scale — the chart shows less detail because each chart unit covers more Earth distance. A smaller denominator means a larger scale — more detail. Now, the key relationship on a Mercator chart: scale changes with latitude. The Mercator is a cylindrical projection, and to keep it conformal — that is, to preserve shapes and angles — the scale must expand as you move away from the Equator. So at the Equator the scale is at its smallest denominator, and as you go toward the poles, the denominator gets smaller — the scale gets larger. Conversely, moving toward the Equator, the scale contracts — the denominator gets larger. Let’s apply that to Example 3. On a Mercator chart, the scale at 54°S is 1:2,000,000. What is the scale at 25°N? The options are: a) 1:2,000,000, b) 1:3,084,000, c) 1:1,121,000, d) 1:3,825,000. First, use your knowledge to discount answers a and c. At 25°N, you are closer to the Equator than at 54°S. Therefore, scale must contract between 54°S and 25°N — the denominator at 25°N must be larger than at 54°S. So the denominator must be bigger than 2,000,000. That immediately rules out a, which is the same 2,000,000, and c, which is 1,121,000 — smaller than 2,000,000. So we’re left with b and d. In some cases, there may only be one appropriate answer given, and the answer can be spotted without calculation. But here, both b and d could be correct, so we revert to the simplified formula. Let’s set it up. Let DA be the scale denominator at 25°N — that’s what we’re solving for. Let DB be the scale denominator at 54°S — that’s 2,000,000. So we write: DA / 2,000,000 = cos 25° / cos 54°. Therefore DA = cos 25° × 2,000,000 / cos 54°. Now, cos 25° is about 0.9063, and cos 54° is about 0.5878. Multiply 0.9063 by 2,000,000, you get about 1,812,600. Divide that by 0.5878, and you get approximately 3,083,806. So DA = 3,083,806, which rounds to 1:3,084,000 — answer b. Notice the logic: because 25°N is closer to the Equator than 54°S, the cosine of 25° is larger than the cosine of 54°. That larger cosine in the numerator pushes the denominator up — the scale contracts, exactly as we predicted. Now let’s move to Example 4, which asks a slightly different question: given a fixed chart distance between meridians, find the Mercator scale at a specific latitude. The problem: On a Mercator chart, the chart length between two meridians, 160°E and 160°W, is 30 cm at 30°S. What is the scale of the chart at 30°S? There are two ways of doing this, and there is very little to choose between them. The first way is to treat it as a departure problem. Scale is defined as Chart Length divided by Earth Distance. So: Scale = Chart Length / Earth Distance. Chart Length = 30 cm. Earth Distance = Departure. And departure is calculated as: Departure = change of longitude in minutes × cos latitude. So here, the change of longitude between 160°E and 160°W — going the short way across the Pacific — is 40°. In minutes, that’s 40 × 60 = 2,400 minutes. Multiply by cos 30°, which is 0.866. So Departure = 2,400 × 0.866 = 2,078 NM. So the Earth distance is 2,078 nautical miles. Now, to get the scale, we need both distances in the same units. The chart length is 30 cm, and the Earth distance is 2,078 NM. To express the scale as 1:denominator, you’d convert the 2,078 NM into centimetres — 1 nautical mile is 185,200 cm — then divide the Earth distance in cm by the chart length of 30 cm. That gives you the denominator. So the scale at 30°S would be 1 : (2,078 × 185,200 / 30). Let me just compute that: 2,078 × 185,200 is about 384,845,600. Divide by 30 gives approximately 12,828,187. So the scale at 30°S is about 1:12,828,000. That’s the first method — treating it as a departure problem. The second method, which the book says there’s very little to choose between, would involve using the scale formula directly with the known chart distance and the Earth distance computed from the meridian spacing. But the departure approach is the one laid out here, and it gives you the scale at the specific latitude of 30°S. So the key takeaway from this page: the formula DenominatorA / DenominatorB = cos A / cos B lets you compare scales at two latitudes on a Mercator chart, and when you’re given a chart distance between meridians, you can find the scale at a latitude by computing the departure — the Earth distance — and dividing the chart length by it. That’s the whole of this section.