Click here for the main/introduction page for the Flat Earth topic.

##### Intro

This must be one of the most persuasive of the Flat Earth arguments. Flat Earthers have come up with ways to disprove the Earth being a globe and prove in stead that the Earth is flat. But is this true? Let us first take a look at their most convincing experiments:

###### Chicago skyline

The problem in the video above is easy to understand. If the globe Earth model is correct, we should not be able to see Chicago from Michigan because it would be hidden below the horizon. If the flat Earth model is correct, we should be able to see Chicago, given the right weather conditions of course. Since the video above proves we can see Chicago from such distance and it is not a mirage, this appears to be a victory for the Flat Earthers. But is that correct? Let’s have a closer look at the method:

###### Bedford Level experiment

The Bedford Level experiment is often cited by the Flat Earther. The Rowbotham method was used to determine the drop of the Earth over distance.

Wikipedia link: https://en.wikipedia.org/wiki/Bedford_Level_experiment

The idea is simple: If you move further outwards, the surface of the Earth drops:

As listed on this table, moving 10 miles outwards would result in a 66 feet drop:

Statute Miles Away | Maths | = Drop |
---|---|---|

1 | 1 x 1 x 8 = | 8 Inches |

2 | 2 x 2 x 8 = | 32 Inches |

3 | 3 x 3 x 8 / 12 = | 6 Feet |

4 | 4 x 4 x 8 / 12 = | 10 Feet |

5 | 5 x 5 x 8 / 12 = | 16 Feet |

6 | 6 x 6 x 8 / 12 = | 24 Feet |

7 | 7 x 7 x 8 / 12 = | 32 Feet |

8 | 8 x 8 x 8 / 12 = | 42 Feet |

9 | 9 x 9 x 8 / 12 = | 54 Feet |

10 | 10 x 10 x 8 / 12 = | 66 Feet |

I would expect the problem with this approach would be obvious to anyone, but it turns out it isn’t. So let me explain:

When are a looking at a distant object located on the globe, we are not looking in the direction parallel to the globe, but a bit downwards, directly at the object:

This means we are not looking at the actual drop of the Earth’s surface between the spectator and the building, but we need to look at the height of the curvature between them:

This obstacle height determines how much we can see of the target. I could give you one formula to calculate this height, but given the bad experience with the method the Flat Earthers use, I’ll take you through the calculations step by step so it can be understood.

###### The mathematics

To calculate the obstacle height, we need to be at the middle between the spectator and target and in stead of the Earth’s radius calculate the distance from the center of the globe to this flat line. Then, the difference between the Earth’s radius and the distance to this flat line will give us the obstacle height.

Step 1

Determine angle α. Here we are cutting a piece from the globe pie. The volumetric mean radius of the Earth is according to Nasa 6371km (3958 miles), which will give us a circumference of 40030km (24873 miles). We arrive at this number with the equation 2 * Π * r

Angle α is then given by the equation α = c / 360 * d, were c is the circumference of the Earth and d is the distance between point A and B, as measured over the curvature of the globe.

Step 2

Determine distance D to the flat line between object A and B. For this we need to know the angle at corner C (center of the globe) between r and D. Since line D is in the middle between point A and B, the angle we need is α/2. We also know angle γ is 90° because it is perpendicular to the flat line between point A and B.

Now distance D can be calculated with the equation cos(α / 2) * r.

Step 3

Determine height H by subtracting distance D from the radius of the Earth: H = r – D.

###### Calculation tool

To help out those who have trouble doing math I have created the tool below to perform the calculation for you. You can use this to easily verify claims made by Flat Earthers, or use it to perform your own experiments. If you don’t trust the results from my script, please do the math yourself.

###### Testing the claims

Now it is time to test the claims! We can look at some of the FE experiments and verify the outcome ourselves. We can even predict the result before we see the outcome and see if we’re right!

Let’s first look at the Chicago ‘mirage’ footage because it is perfect for this purpose.

The original photo was taken from across Lake Michigan from the Michigan shore (to be more precise the location was Indiana Dunes) at a distance of approx. 40 miles away. Some websites mention the distance was almost 60 miles, but measuring the distance in Google Earth gives use approx. 35 to 40 miles, so let’s go by 40 miles. By the logic of the Bedford level experiment the drop would be 40 * 40 * 8 / 12 = 1066 foot or 0.2 mile / 324 meter. The highest building in Chicago is the Willis tower with a height of 1450ft (442m). This means it should be impossible to see Chicago from this distance, or at most a small portion of the highest building would show.

Now, let’s see what my calculation says about this. When we enter the distance of 40 miles it will result in an obstacle height of 266ft (81m). This means we should be able to see most of the high buildings. Note that at point B we need at least twice the obstacle height in order to see anything.

What we now expect to see is at maximum the area between the yellow line and the upper red line.

Compare this to the photograph from Joshua Nowicki:

As you can see, Joshua’s image looks just like we would expect from such a distance if we are in fact on a spherical Earth.

If the Earth was indeed flat, we should have seen more of the skyline. To be more precise, we should see the buildings from the ground to the rooftops. To visualize this I have blended the close up image into Joshua’s photograph so you can see how much is hiding below the horizon. Please note that both images are taken from a different angle, so they do not line up horizontally.

My conclusion: This is absolute proof the curvature of the Earth lines up with the globe model we are all familiar with. Despite the fact Flat Earthers like to use this photograph to proof the flat Earth model, it actually disproves it!

but wait…

###### Luuk, you’re all wrong!

What some of you are likely to point out is that the commonly used flat Earth math is not so wrong after all. If we are so close to the surface of the Earth, the line of sight would be tangent to this surface and therefore level. So, the claim would be we can not look directly (down) at the distant target, but in stead the line of sight would always be level.

My response to this: This is only true if the eye is at ground level. Since the diameter of the Earth is so incredibly large, this tangent effect is hardly noticeable. If we move the point of view just a little bit up we no longer only see level, but we can look down over a very large distance. To proof my point I made the following video to demonstrate this effect:

You can easily test this for yourself. Take a telescope or good binoculars and look over the water to very distant object, such as the Chicago skyline over Lake Michigan. When the object is in view, slowly move the camera down and you’ll see you’ll have to come close to the surface to make the object disappear. This also demonstrates the curvature of the Earth.

Please feel free to comment/criticize below!