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.


  1. I see you saying the math is incorrect; what is the correct formula? I don't need to do 1000s of miles. Just an approximation for 1 to 100 miles. How close is 8 x 100 x 100 = 80,000?

    1. To be brutally honest, it's more than a little annoying to have you asking this question when I provided a link to the math in the body of my post. At 100 miles, the 8m^2 inches formula would already be way off, and as I said, might be answering the wrong question in the first place. Please read the blog post again.

    2. The 'correct' formula depends on what you ACTUALLY want to measure.

      8"xmiles^2 is a fair approximation of the DROP HEIGHT as long as miles is under about 100 miles -- after that it actually (and increasingly) underestimates the DROP HEIGHT.

      HINT: 8" x miles^2 is a parabola, not a sphere.

      But Gordon's point is that, unless your eyeball is LITERALLY half under water (center of the lens) then DROP HEIGHT is *NOT* the correct thing to measure -- what you are supposedly asking is "how much of a distant thing is hidden to MY EYE".

      1+1=2 but that equation doesn't tell you much less than 8" x miles^2 does about This Specific Question.

      Understand now?

      So... what formula to use?

      Did you miss Gordon's link to the calculator?

      If you want more details on the map I also did a blog post on it:

      And I have my own calculator which shows the formulas:

      And this guy wrote a REALLY cool rendering engine that compares Flat vs Globe:

    3. I have a question,, if i stand in the Water... and put my telescope 2 feet above the Water and I have a friend 400 miles away and he stands on an island in the Water with a Rod that has a target 2 feet above the Water and i look in my Telescope would I be able to see his target 2 feet above the water if the earth Curves?.. and if at 400 miles away 2 feet above the water on both ends thru a high power 10 inch telescope and i can see the target.. Why?... Just a Question on something we tried... Please give me help me out why I can see the target?

    4. Somric C No, you would not be able to see the target. Not 400 miles away, or even 100 miles away. It's telling that flat-Earth videos on this subject are always at marginal distances. On a flat Earth, it should be possible using only binoculars to see the Sierra-Nevada mountain range from atop Mauna Kea in Hawaii, 2500 miles away. But, alas, it is not, because the Earth curves.

    5. Hi, I live in Mauritius island in the indian ocean right next to Reunion island with 250kms separating the two islands. Reunion island has an active volcano on it. How is possible that on a clear nights, glare from volcanic activities in Reunion can be seen from the Mauritius coast? According to the calculator provided in the blog post, the two islands are separated by a 4000m elevation? Plus on clear winter days when the atmosphere is free of particles, any island can be seen from the other by using a pair of binoculars only? Photos from powerful cameras have also been taken? In fact it is such common knowledge that people dont even think about curvature, they are only amazed that they can see where their cousins live, hundreds of miles away

    6. ad pat: You'd have to show me the photographs. The SW shore of Mauritius is 112 miles from the NW shore of Reunion Island, and it's about 128 miles to the peak of Piton des Neiges. If you are actually standing on the water, the mountain should be hidden, but not necessarily the glow from the volcano, especially if there are clouds.

      I'm skeptical, though, of seeing Mauritius from the shore of Reunion. From the peak of Piton des Neiges? No problem. But anything else, I'd need to see photographs with verified data (including longitude, latitude, elevation, direction, and weather) in order to evaluate them.

    7. LOL, complete Garbage. You CAN see a ship over 400 miles away with a binocular. With the naked eye, it looks like the ship is sinking. But this is due to PERSPECTIVE.

      The further way the objects go the smaller it is and to the naked it, it looks like it's gone and under water. But all you have to do is grab a binocular and the ship magically reappears. Flat Earth!

    8. "Admin" That's complete rubbish. Prove it. You saying "You CAN" is meaningless. Come up with proof and stop parroting the stupidity of flat-Earthers.

    9. Here is another example. Buoy disappear into the ocean. Proof earth is Round! Wait, zoom in and the buoy appears again. LOL!

    10. "Admin," I'm pretty much done with you if this is the extent of your contribution. The buoy is not beyond the horizon, and it's obvious that it is not. The is no verification of the distance from the photographer to the buoy. Stop being ignorant, or keep it to yourself.

    11. And I'm not publishing any more comment with nothing to offer but links to YouTube videos we've all seen and debunked ad nauseam, nor insistence on proof of a globe (from anyone who's obviously not read the blog).

    12. The Math is clear, unfortunately flat earthers are using wrong math and claim it's right.

  2. I wanted an easy calculator to return to that would allow me to calculate the bulge over a length of curvature. Mainly cos Jeran's always using 8inch formula wrongly to calculate it.

    Well I learned it's called a sagitta for a start. The distance from the centre of an arc to the centre of it's base.

    Anyway I ened up with this:

    1. It looks good to me. The only problem I see, and this is the flat-Earther's problem and not yours, is that they will still expect that this is the amount of something that should be hidden, neglecting that line segment i still meets the surface of the Earth at a viewing altitude of 0.

      But the more information the better!

  3. Flatearthers know about observer height and include this in their calculations. Please tell me again how you get the figure 7,950 as the diameter of the earth? Google states it is 12, 742, are Flatearthers in charge of Google now? Where did your 12,500 come from? That's two numbers you've gotten wrong, pretty sloppy for a math geek.
    Flatearthers know that the eight inches per mile squared isn't perfectly accurate over uber distances, it is accurate for making empirical observations.
    Flatearthers know about refraction and perspective, but obviously you do not; it's perspective that causes the sun to set, not refraction, and certainly not curvature. How much refraction (due to humidity) can there be above a desert, or over a frozen lake? Lasers. Videos. Proven. No curve.

    1. The mean diameter is actually 7,914 miles, or 12,742 kilometers. Know the difference between a mile and a kilometer? Hmm. 12,500 miles is half the approximate mean circumference of the Earth. A closer approximation would be 12,450 miles.

      You want to split hairs? No, no you don't, because that will definitely lead us away from every "proof" you've ever seen of a flat Earth.

      And no, many, many flat-Earthers do NOT include observer height in their calculations, which is why I wrote this post.

      And no, perspective cannot cause the sun to set. If you think it can, then all of your knowledge of perspective comes from flat-Earthers, and none from any practical application.

      How much refraction can there be over a desert or a frozen lake? Depends on temperatures. There can be quite a lot.

      And as for your videos and lasers that supposedly prove no curve, I'll take the work of experienced geodetic surveyors over your YouTube dweebs any time.

      Why don't you stop watching YouTube and get out in the field and do some real observations? Use you head and stop letting incompetents do your thinking for you.

    2. There are so many proofs of the flat earth, we don't need to argue your Masonic mathemstics bs.
      Yes most flat earthers DO account for observation height in their experiments, its too bad some of the globalists dont. Such as on a recent episode of genius, when Stephen Hawking's body double tries to prove curvature over a lake, using a laser. They neglect to take the laser height into effect, and ironically the results they claim at the distances they measured are exactly the results of the 8" /m^2 formula. "Stephen Hawking fakes experiments to prove the globe", by Jeranism.

      Yes; perspective is how the sun sets, the sky meets the ground at the vanishing point of our vision, objects will shrink and disappear the farther away they are. Planes appear and disappear at the horizon yet they are still thousands of feet in altitude. Just as the sun and moon do.

      Refraction due to atmosphere is a bogus theory, just a band aid applied to stem the bleeding of your dying globe theory. The atmosplane actually magnifies distant objects when conditions are right, this is called looming. Often these magnified objects are partially obscured due to, yes, Perspective.

      As for the YouTube dweebs I watch, they are critical thinkers and their videos are eye opening. I will trust their word long before I trust some jar head empty vessel, government shill that simply parrots what he's told.

      I have lived for many years beside an ocean, as well as a large lake, and I have seen islands, shorelines and cities many miles distant that should have been invisible or at least partially obscured.
      I have stopped drinking the kool aid and the flouride as well, and have given a lot of thought to this topic. There is no proof of the globe.

    3. The fact that you even use the phrase "Masonic math" shows your complete lack of critical thinking skill. I wouldn't even bother to answer you, were it not for the fear that someone might stumble upon this and actually think that you have a clue as to what you are talking about.

      Genius. Hmm. Now why do you supposed the height of the laser wasn't taken into account in that experiment? Oh, maybe it's because they weren't measuring the line of sight, usually expressed as the hidden height, but the drop from one distance to the next which, lo and behold, is exactly what 8*m^2 inches is meant to estimate.

      And referring to any work by Jeran Campbell as a source destroys your credibility. He very simply has no idea what he's talking about, and doesn't even remember what he's said. He claimed, on a comment on this blog which he never bothered to delete, that he the "truth" of the flat Earth would be revealed in December of last year. I'm still waiting.

      Refraction is real, even Rowbotham knew it, but he chose to ignore it, knowing full well what the results would be. Looming isn't due to perspective, it's due to refraction, which is what causes mirages in the first place. Get your head out of the flat-Earth cloud and go learn something real.

      And I don't care how long you've lived by the water. If you think that perspective can explain sunsets, then you just don't understand perspective at all. Until you can quantify for me how a sun that's (by any calculation) 108 times as far away as it is across can appear to sink below the horizon before it disappears in the distance, or what that distance will be (in a way that can be reconciled with a flat plane) you're nothing more than a blowhard sheep.

    4. If the Earth was really a globe and Astro-NOT would have taken a cell phone picture by now. The author has never been to space or seen the Earth from space, and every single photo of Earth is Composite = faked. The photos are what Rob Simmons came up with from numbers and data. Plus WTH haven't we been back to the moon? Made a base? We're going to Mars but we can't make a base on the moon? Makes perfect sense. Never mind the Satanic Science of Earths Orbit of the sun 66,6000 mph and Axis 66.6 deg If you think those numbers are coincidence you must be a coincidence theorist.

    5. It took me several hours to decide whether or not to publish the reply from unknown, but I went ahead so that I could deconstruct a little flat-Earth thinking, or the lack thereof. Before we start let me mention that, once again, this comment is completely off-topic.

      So let's start with the astronaut taking a cell phone picture of Earth. Astronauts have taken lots of pictures of the Earth, as I have already written about, but any that could have conceivably been taken with a cell phone would have been taken from less than 500 miles away. Let's do a little math:

      The Earth is about 7950 miles in diameter. So 500 miles (and I'm being generous) is a little more than 0.0625 Earth diameters away. Let's use a basketball as an analog. A basketball is 9.5 inches in diameter. A camera in low-basketball orbit would be about 0.6 inches away from the basketball. Now, if the person making this comment can show me a picture of an entire basketball, taken with a cell phone, from .6 inches away, then we can talk.

      On the second point, it is true, I have never been to space. Nor had anyone else until 1961, and yet humanity knew the Earth was a globe long before that. Your point is pointless.

      Now to the trope about every photo of Earth being fake, Robert Simmon, yada yada. I've already written about this elsewhere, it's a load of crap, so unless you can come up with something real, just shut up.

      Why haven't we gone back to the moon? Well, money, and a lack of public support, plus some really lousy timing with Watergate, the Vietnam War, and the hyper-inflationary 70s. It really has little to do with technology. As far as Mars, that was the real target all along, and I'm glad to see some people finally acknowledging that.

      And to the last point about these "coincidental" numbers, I make two counterpoints: 1) You have to do some rounding to get those 666s to show up, and that's cheating, and 2) The Biblical Mark of the Beast is not the digits 666, but the actual quantity Six hundred and sixty-six, as anyone could figure out with a little bit of thought about the Greek numbering system at the time of the authoring of the New Testament (which was, if you didn't know, written in Greek). So you even fail on theological grounds.

      In other words, as usual, the flat-Earthers dredge up the same old garbage and never come up with anything remotely useful.

      Or even relevant.

    6. just curios about the moon ... is there any film footage of them doing a full 360 pan of them standing on the moon or anywhere in the flight?... if so.. can you give me a link.. being in the film industry with all the footage of the landing and walking, driving and doing other things like golf etc... i can't see that it's possible not to happen... in the film industry ... you never show the 4th wall... and i know.. since we went.. this would help me believe more.. if you can give me a link of a 360 degree pan of the moon shot by one of the astronauts... thanks for helping me with this..

    7. I do wish you'd make these comments on a post that's actually about the moon landing; it would help keep things straight. The Apollo astronauts did not shoot film on the moon; they shot lots of stills, and the video was broadcast live and recorded on Earth. It might be that there is such a pan in the remote-pan-head footage from missions 15, 16, or 17, but I don't really need to go look. Aside from having witnessed the broadcasts live, and having then studied film, I know that there are many aspects of the moon footage that could not have been faked in the 1960s and 70s. If you want to go looking, be my guest.

  4. Distance = 4 Miles (21120 Feet), View Height = 6 Feet (72 Inches) Radius = 3959 Miles (20903520 Feet)
    Horizon = 3 Miles (15838 Feet)
    Bulge = 2.67 Feet (32.01 Inches)
    Drop = 10.67 Feet (128.03 Inches)
    Hidden= 0.67 Feet (8.01 Inches)

    With Standard Refraction 7/6*r, radius = 4618.83 Miles (24387440 Feet)
    Refracted Horizon = 3.24 Miles (17107 Feet)
    Refracted Drop= 9.15 Feet (109.74 Inches)
    Refracted Hidden= 0.33 Feet (3.96 Inches)

    Tilt Angle = 0.058 Degrees, (0.0010 Radians)
    Horizon Dip Angle = 0.043 Degrees, (0.0008 Radians)

    Horizon Curve Fraction = 0.00010
    Horizon Curve Pixels = 0.33
    Horizon Curve Angle v1= 0.00672
    Horizon Curve Angle v2 = 0.00672

  5. The most consistent attribute of FE-dom is the determned deliberate failure to develop even the most rudimentary alternative model of solar movement mechanics - including solar diameter, distance, mass, etc. which better explain the observable day/night, summer/winter phenomena than the current "spheroid rotating about an inclined axis orbiting in an elliptical orbit about a huge sun at a huge distance" model...

    By now the FE-super-geniuses could reasonably be expected to have produced an open source 3D computer simulation model with equations and physics and stuff of their flat-disc-at-absolute-rest-in-the-centre-of-the-universe concept.

    For brownie points:
    Add lunar dynamics... including phases.
    How thick is the disc-earth?
    What is on the "bottom" ? or does everything fall off cause there is no gravity ?
    What is the temperature of the dark-side-of-the-earth?
    Ahhhhh... is that why Antarctica is cold? Bleed up from the dark side?
    Maybe there is a hole in the centre too to bleed the cold up into the arctic?

    For full-meal-deal-black-hash-brownie points, include an alt-planetary-"orbital"-mechanics model which better accounts for the observational data.
    How about auroras ?

    How about more math and physics and less wankering about phenomena beyond your perceptual comprehension...
    Give the little head a break.

  6. Can what I have mentioned be considered practical application on the field?

  7. I really admire and respect the fact that your opinion is so contrary to Flat Earth thinking. That takes guts and confidence in your stance.
    However, if you use just human instincts,intuition, and simple observation, it is quite obvious that the sun and moon are very near to the earth. The Bible is not wrong. Genesis is quite clear. The earth does not move etc...One must look at all the possibilities from both sides and make their own opinion. This is very analogous to the controversy between creation vs evolution. Or God vs the devil.
    It comes down to your beliefs and faith. Science is always searching but never finding the truth.

    1. Well, once again a comment which is completely unrelated to the post. Oh, well. Look, the one thing that makes itself most obvious to anyone who actually observes the sun and moon is that they are not close to the Earth. The fact that their apparent size stays nearly the same throughout the day and over the course of weeks is enough to cinch that fact.

      And that fact is enough to destroy the notion of a flat Earth. Case closed.

      And, excuse me, "God vs the devil"? Really? We're talking about simple facts here, not some great moral dichotomy. Get off the pulpit and get out into the world and see what's around you.

      And read the blog so that you make comments related to the topic of the post. Please.

  8. 1,250,000,000 inches is not 236,742 miles. You needed to convert inches to feet before you divide by 5280 or just divide by 63360 to give 19,729 miles. Flatards are too dumb to have seen this. We publish our ideas so they may be scrutinized, corrected/agreed with, so we all get a little smarter. There is no NASA cover up so the government can take your money (NASA only formed in 1958) Go look at how money is created, there is no need to steal it.

    Just because you flatards don't understand something it doesn't mean it's false. Either keep trying to understand, ask for help, or just agree that the people being paid to do their jobs are doing them and when they give an answer to your question...accept it.

    Nice post Gordon Brooks. I hope you have helped answer a question or 2 for those flatards that are trying to understand.

    1. Oops! You're right. I stand corrected.

  9. Thank you for this blog, I must admit Im not ready to go full FE but they have brought forth some very disturbing theories, mainly about the many space agencies stealing our tax money with clearly fabricated images and numerous contradictory stances. Its given them more credibility than I would have normally given but Im not done searching. Thanks again

    1. What do you think the contradictions are? And why do you think that space agency images (aside from the ones they clearly label as visualizations) are fabricated? Aside from flat-Earth and moon-hoax claims, what evidence have you really investigated? I'm genuinely curious.

    2. How about the fact that the NASA "Live feed" of Earth from the International Space Station is completely CGI.

      If you even driven in a car, or flew in a plane, you would know that objects from a distance "look" to move slower than objects that are closer.

      Yet, the CGI image of the ISS flying over earth shows all objects moving at the same speed. Which is how CGI video works.

      If you fly on a plane, objects from a distance move slow, but as you fly under it, it moves really fast. You don't see that in the fake CGI ISS flyover of the Earth.

    3. Normally I wouldn't respond to someone who posts with a hidden profile, but "Admin" has made two claims that are easily testable. The first is that the feed from ISS shows all objects moving at the same speed. I have actually measured this in response to Dave Murphy making the same claim. My findings, copied from another blog post:

      A look at the footage he supplied shows that an easily-identifiable cloud feature moved eight pixels from one frame to the next when at the top of the globe, and 57 pixels from one frame to the next near the bottom. Case closed, Dave, you're just wrong.

      The other claim is that CGI would show objects moving all at the same speed. But any 3D program I've ever encountered takes perspective fully into account, so I've never seen this alleged lack of speed variation between near and far objects in computer-generated animation.

      Clearly this poster has been watching way too many flat-Earth videos, likely including Dave Murphy's sorry excuse, and should spend more time actually observing and measuring things.

  10. Thank you!!! I'm so tired of asking flat earthers where they got that formula because I knew it couldn't be accurate.

  11. Great answer in your blog.
    Unfortunately it needs to even more simplified for flat earthers to understand, going by some of the rebuttals.