Thursday, December 22, 2016

Eight Inches Per Mile Squared

It seems to be the only math which many flat-Earthers are willing to consider accurate. The Earth, they say, supposedly curves away at a rate of eight inches times the distance in miles squared. Which is true. Kind of. But not really.

The figure, which they say comes from NASA, or "science," actually comes from a very different source. Flat-Earthers, no matter where they got it themselves, owe it to none other than Samuel Birley Rowbotham, author of Zetetic Astronomy. He got it from the Encyclopedia Brittanica, where it is cited under the heading "Leveling." You'll find his lengthy quote (I doubt that he got permission to use it, by the way) starting on page 8 of the 1865 edition of his book.

The problem is that this is in the context of civil engineering, not mathematics, and it's just a rule of thumb employed by plane surveyors to compensate for the drop in a target of the same height as the surveyor's transit. It builds up inaccuracy as the distance increases for two reasons, the first being that it is not exact, and the second being that it is not based on the formula for a circle. It actually plots out to be a parabola.

Don't believe me? Let's take an extreme example, the most extreme possible in the globe model, in fact. So, what is the farthest apart two points can be on Earth? The answer is about 12,500 miles apart, because if they are any farther, they will be closer from another direction. What is the maximum drop a target can have? About 7950 miles, the diameter of the entire Earth (although, looked at another way, the drop at 12,500 miles is zero). How does the formula stack up?

The square of 12,500 is 156,250,000 and 156,250,000 times 8 inches is 1,250,000,000 inches, or 236,742 miles. About the distance to the moon. See what I mean? Eight inches per mile squared never curves back on itself, and becomes progressively less circular as the distance increases.

But that isn't even the biggest problem with eight inches per miles squared. The biggest problem is that flat-Earthers, with few exceptions, apply the formula to the wrong situation. They apply it to the question: how much of a distant building or land feature should I be able to see from a given distance? And for that question, eight inches per mile squared is useless, unless you want to posit that every observer is lying down with his face planted firmly on the ground with one eye closed.

The simple fact is that you can't answer this question without taking at least two things into account. The first is the height of the observer. I won't go into the math, because not only is there an excellent online calculator for this, but the math and source code are given for it right here.

The other thing you have to take into account is atmospheric refraction, the fact the different densities of fluids such as air bend light. Flat-Earthers love to invoke refraction to explain sunsets, as much as they like to disparage it as an excuse when viewing distant objects.

For just a little information on how refraction can skew observations, especially close to the ground, and especially over water, I refer you to none other than page 10 of the 1865 edition of Zetetic Astonomy.

Puts a whole new light, I think, on Rowbotham's experiment at the Old Bedford Level.