Extraordinary claims require extraordinary evidence.

## Introduction

It is Easter and I have a terrible cold − I am not going to see family this year. Instead, I am going to lay here and write for a while.

Yesterday, I received a question from a reader who was puzzled by a web page written by a flat earther that presented a seemingly rational argument in favor of the flat earth position. In a nutshell, the flat earther's argument says that to see a lighthouse at long distance on spherical Earth would mean that you would have to be able to see around the horizon, which they claim is not possible. Therefore, the Earth must be flat. I may be mischaracterizing their argument, so you may want to visit web sites that go into the details of the flat earth rationale. Of course, I argue that refraction can and does literally allow you to see "around" the horizon.

The questioner's request was to help him understand the fallacies in the arguments being made. I generally avoid these discussions and simply refer people who write me on this subject to wikis and blogs that are focused on these topics. For example, in the case of the flat earthers, the RationalWiki does a good job of debunking their arguments. However, I am in bed and feeling cranky, so here I go.

While I will not argue about non-falsifiable arguments, I can discuss the parameters critical to computing lighthouse ranges and how nominal lighthouse ranges are computed in a standard table of lighthouse ranges. As a side topic in Appendix A, I review one of flat earther calculations and with my own calculation show that refraction and tidal water level changes can easily explain the ability to see a lighthouse over-the-horizon. You do not need to assume that the Earth is flat.

## Background

I have done a fair amount of optical range modeling in the Earth's atmosphere. Truth be told, this is all related to my interest in battleships and optical fire control systems. However, I did write a blog post that referenced lighthouses and the material presented here is based on that post.

## Analysis

### Formula

#### Starting Point

Equation 1 gives the distance one can see from an elevated point (e.g. lighthouse or battleship mast) to a point on the horizon, assuming a nominal level of refraction.

Eq. 1 |

where

*s*is the distance along the Earth in kilometers.*h*is the height above the sea in meters.

The weakness of this formula is that the front coefficient (3.86) assumes a specific lapse rate, which describes the atmospheric temperature variation with height. Variations in the lapse rate can produce enormous increases or decreases in the visual range of a lighthouse. Variations in the lapse coefficient can even explain atmospheric ducts that can guide optical signals (and other electromagnetic signals) over very long ranges.

Also, the height of the lighthouse above the sea varies with time (i.e. tides) and weather (e.g. storm surge). Most official tables of lighthouse ranges are based on the mean high water level about the lighthouse.

#### Model Representation

Figure 2 shows how the visual range of a lighthouse depends on both the height of the lighthouse and the viewer. Figure 2 is actually taken from a post on battleships, but the geometry is identical for both cases. This means that I must apply Equation 1 twice:

- Compute the range from the lighthouse to the horizon.
- Compute the range from the viewer to the horizon.
- Add the ranges together.

#### Lighthouse Formula

Figure 3 shows my lighthouse formula. I also grabbed some data from an old UK publication on nominal lighthouse viewing ranges for comparison.

This formula also reveals another source of variation − the height of the observer above the sea. This will vary since the height of the deck of each ship is different. The standard for computation is 15 feet, but that can vary dramatically by the type of ship. For example, the flat earther web page in question here assumed a 24-foot high deck. To duplicate the UK table values, I needed to use 15 feet for the deck height.

### Results

Figure 4 shows that Equation 1 from my blog post produces the same result as that specified for nominal lighthouse ranges. So I believe that my model is reasonable.

## Conclusion

Reported lighthouse ranges are a strongly dependent of the assumptions made in their calculation. You do not need to assume the Earth is flat to explain how lighthouse range estimates can vary so much from a pure spherical model. The variation naturally occurs because of changes in water level and temperature.

## Appendix A: Worked Flat Earther Example

I will work one of the flat earther examples. My intent is to first show how they performed their calculation. I have highlighted the statement "not an uncommon thing", which is a very imprecise statement. In general, lighthouse ranges are stated in a conservative manner that ensures that they are almost always visible at the stated range. This statement indicates that the ranges quoted here are NOT common.

The distance across St. George's Channel, between Holyhead and Kingstown Harbour, near Dublin, is at least 60 statute miles. It is not an uncommon thing for passengers to notice, when in, and for a considerable distance beyond the centre of the Channel, the Light on Holyhead Pier, and the Poolbeg Light in Dublin Bay. The Lighthouse on Holyhead Pier shows a red light at an elevation of 44 feet above high water; and the Poolbeg Lighthouse exhibits two bright lights at an altitude of 68 feet; so that a vessel in the middle of the Channel would be 30 miles from each light; and allowing the observer to be on deck, and 24 feet above the water, the horizon on a globe would be 6 miles away. Deducting 6 miles from 30, the distance from the horizon to Holyhead, on the one hand, and to Dublin Bay on the other, would be 24 miles. The square of 24, multiplied by 8 inches, shows a declination of 384 feet. The altitude of the lights in Poolbeg Lighthouse is 68 feet; and of the red light on Holyhead Pier, 44 feet. Hence, if the earth were a globe, the former would always be 316 feet and the latter 340 feet below the horizon!

Figure 5 shows how I duplicated the flat earther's calculations. There are some issues with their calculation:

- purely geometric, no refraction.
- assumes a 44-foot Hollyhead lighthouse (Wikipedia states its height as 70-feet ) and a 68-foot Poolbeg lighthouse (reference states 66-feet )
- Lighthouse heights are normally stated with respect to the high-water level, but I am unclear what the flat earthers are using.

Figure 6 shows the geometry that I will be assuming here.

Figure 7 shows that the lights could easily seen at low tide and with reasonable atmospheric conditions.

I was curious as to what effect on the value would be if I played with the 3.86 km constant in the above formulas. It turns out that 3.85 provides a better match to the Specified Range values. The best fit, ignoring significant digits, was 3.84783759. These values were found by Excel Solver with this criteria:

1. Minimize Sqrt(Sum Diff^2)

2. By changing the 3.86 factor

The leading coefficient is primarily a function of the lapse rate and everyone assumes a different value. When I do my calculations, I try to use NASA or the Wikipedia (i.e. sources readily obtainable on the web). However, there are numerous different sources and they all have different values. In the case of lighthouses, the sources are old and we may never know what exactly they used. Your approach to determining the coefficients is probably the best way to go today.

The solver add-in for Excel is an excellent way to optimize coefficients. I use it myself for many things.

mathscinotes

earth IS flat- employ: a gyroscope after becoming familiar with its rigidity in space; a laser beamed over a still lake after dark; sunlight splayed through patchy clouds from "93MILLION miles away" ( REALLY? common sense doesn't kick in and reason that impossible as rays would ALL BE PARALLEL!!)

Flat Earther main tactic: Unable to address any specific points raised by an article, so go for some flat Earther goon-approved claims to distract people...

Same old fallacies we see debunked over and over.

Gyros are adjusted by schuler tuning for navigation.

"Lasers beamed" are subject to atmospheric refraction close to a surface like ALL light.

The sun's rays ARE almost parallel. Remember when you see rays and shadows under clouds, you are looking up at the sun and perspective makes the rays converge at the sun. Evidence: Look for images of the sun's rays see from the side over clouds to see them parallel.

Evidence: Stand at the base of some skyscrapers for the same effect.

And really, don't appeal to "common sense" because you didn't use any actual confirmable evidence... Try using evidence.

And it's simple. How can the sunset light the TOP of any object or clouds and not the base or you on a flat Earth with no land based obstructions.

I’ve always been curious to determine the visibility of a light from a given elevation that negates earth’s curvature.

No one wants to be trapped in a snow globe. I very much like the idea of living on a ball earth. When I was younger I wanted to have a space ship. I wanted to live on the moon. However, I want to know the truth and I don't want to be lied to. There are questions that remain that are incompatible with a globe model. While the example you used may be feasible on a ball earth, there are several other examples that are more difficult to rationalize mathematically:

The Isle of Wight lighthouse in England is 180 feet high and can be seen up to 42 miles away, a distance at which modern astronomers say the light should fall 996 feet below line of sight.

The Cape L’Agulhas lighthouse in South Africa is 33 feet high, 238 feet above sea level, and can be seen for over 50 miles. 1,400 feet below an observer’s line of sight.

The Statue of Liberty in New York stands 326 feet above sea level and on a clear day can be seen as far as 60 miles away. 2,074 feet below the horizon.

The lighthouse at Port Said, Egypt, at an elevation of only 60 feet has been seen 58 miles away, where, according to modern astronomy it should be 2,182 feet below the line of sight.

Another example is the Notre Dame Antwerp spire standing 403 feet high from the foot of the tower with Strasburg measuring 468 feet above sea level. With the aid of a telescope, ships can be distinguished on the horizon and captains claim they can see the cathedral spire from an amazing 150 miles away. If the Earth were a globe, however, at that distance the spire should be an entire mile, 5,280 feet below the horizon.

I harvested these examples from this site: https://aplanetruth.info/2015/03/29/22-is-the-earth-a-sphere-lighthouses-and-distant-lands/

These were several examples but I found it difficult to locate verification of these claims. There was one, however, that was substantially verifiable. Joshua Nowicki took a picture of the Chicago skyline across Lake Michigan, 60 miles away. There should be 2400 ft of declination at this distance. The tallest building in Chicago is Willis Tower at 1451 ft.

https://ourwayisthehighway.files.wordpress.com/2017/04/chicago_from_michigan_joshua_nowicki_photography.png?w=940

"However, I want to know the truth and I don't want to be lied to"

Yes you do. You want to believe lies that allow you to nurture the fantasy that you can see things other people can't. All these phenomena are perfectly explicable in terms of atmospheric refraction. The above article mentions atmospheric ducting.

https://en.wikipedia.org/wiki/Atmospheric_duct

Sailors, surveyors, signallers and others whose jobs are affected have been accounting for the highly variable refractive properties of the atmosphere for centuries, and none of them have ever ever concluded that this phenomena proved that the Earth was flat (because they actually understood it). You've been fed nonsense on conspiracy channels and told that believing it will prove that you're special. If the Earth were actually flat, there should be no obstructive horizons at all. Viewing distances should be limited only by atmospheric transparency. Objects certainly shouldn't disappear from the bottom up.

Grow up and reject these infantile fantasies and stop swallowing nonsense you're fed by charlatans like Dubay, Jeranism, Rob Skiba and the rest of these liars who are taking advantage of your ignorance and credulity to line their own pockets.

You didn’t actually refute what was stated, you just complained about it

If the Earth is a sphere, why does the horizon rise to meet your eye level, no matter your height? According to trigonometry this shouldn’t happen

The horizon doesn't "rise up to meet your eye level" - and this can be very easily verified for oneself. Would you like to know how to do this?

Mark,

"A Plane Truth" is not a reliable source. I believe he lifted those numbers from Eric Dubay's video. And as you say they are hard to verify. And are not true in almost every case. You really can't see the Statue of Liberty at sixty miles on a clear day. Joshua Nowicki shoots that location from Michigan to Chicago year round. He has presented scores of shots of superior mirages of Chicago at various levels both inverted and not. Even on the clearest days he can not see Chicago at all. But under the right temperature conditions with a temperature inversion layer and at the right time when the sun is below the horizon he has photographed up to 12 Chicago skyscrapers. But says himself in the documentary about him on you tube its rare for him to see more than four of the tallest barely peaking over. Your photographer is a globe earther. So is Pic Gaspard who shoots record long distance shots land to land. He uses globe science to acquire these shots.

If you feel you have been lied to the solution isn't believing a bigger lie to spite the liar. I live in Los Angeles and watched Space Shuttles land here for twenty years a week or two after they launched in Florida. Now a new rocket from Space X or NASA seems to go up every three months on average and is visible from my back porch. For those who claim the rockets get dumped into the sea. You can watch You Tube videos of the rockets as they travel down the coast to Mexico, from Anaheim, Capistrano, San Diego and Baja you can see the rockets are continually climbing until they can not be seen. And people in Phoenix Arizona can see the entire path of the flight from 500 miles away almost as clear as we do on the coast of California. A bright light can be seen for 500 miles or thousands of miles on a globe earth or a flat earth but only if its above the horizon. On a flat earth any bright light source above the sea is high enough to be seen for hundreds of miles. But that just doesn't happen.

Hello your diagrams all show the light from the lighthouses pointing downward towards the water, but anybody who ever looked at a lighthouse knows that would never happen, the light is pointed straight out, not tilted down in anyway. I can view the oldest working lighthouse in the world 15 minutes from me, at Dunmore East co Waterford Ireland, the Lighthouse is on HookHead Co Wexford and its 300 years old, and I have been in it and I was informed by the guide about it, so this pages diagrams are completely misinforming people, or are just plain dishonest....

You're right of course that the light points straight ahead. But his diagrams are also right: they're not showing the lighthouse light, as such, but tracing a path between the observer and the light. In reality, the light is shining in all directions, including down, up, and to the sides. But coloring all that in wouldn't be very useful for the diagram.

Does that make sense?

Actually, what you are asserting doesn’t make sense. A lighthouse light works not by being very strong but rather by orienting the light in a singular direction. Study up on Fresnel lens. The light is sent at a right angle to the vertical orientation of the lighthouse. As such, using the battleship example where sight angles could be oriented other than a right angle to the vertical orientation off the sea level should not be used to respond to questions regarding lighthouses.

It's possible Bob underestimated how badly you misunderstood the diagrams here. They do NOT show "light from the lighthouses pointing downward towards the water." With this claim, you are either being disingenuous, or betraying your lack of experience interpreting diagrams.

The diagram is not to scale. Specifically, vertical scale (relative to the center of the earth) is exaggerated, while circumferential scale (distance) is vastly shrunk. If you made no change to the geometry of the diagram (Figure 6, say) and only corrected for scale, the height of the lighthouse on the far right would still be, say, 30 pixels, while the ship/observer over the horizon (currently in the center of the diagram) would now be positioned something like 60 feet to the left of your monitor, and the line representing the "light beam" would appear to be virtually parallel with the arc of the earth's/ocean's surface.

No difference to the math or the model. The light beam goes straight out at the horizon, and that's what the diagram shows.

Interesting article. Thanks.

A couple of quick comments:

the Carl Sagan quote is somewhat...duplex. The claims with respect to much of standard cosmology are quite extraordinary themselves, and not backed with physical evidence (beyond pointing to the thing itself, as proof OF itself)

terrestrial refraction is just one such example. the...circularity of the 'evidence' is such that refractive indices are relying on a radius value where the radius is not derivable from a geometric, physical horizon. the horizon being variable due TO refraction negates its physicality. there simply isn't a fixed R to the 7/6ths R portion of the math

just something to consider - and if already considered and dismissed without due consideration, then perhaps to reconsider and assign the proper weight to this critical issue of a non-physical horizon and its implications to all that is derived from assumptions about it (even to the point of deriving the sun's distance)

Models and means.