The Earth is a sphere after all..

In the challenge section of this web page, I tried to give easy experiments for anyone (in particular Flat Earthers) to get the necessary evidence supporting a Flat Earth. There are four experiments ordered in increasing difficulty. But suffice it to say: All challenges are impossible, even the simple first one. How can I be so sure about this? Well, I know the Earth is a sphere, and I base this on two simple backyard experiments explained below.

These experiments have some things in common: 

So these two experiments is all you need. They form a Swiss army knife for any discussion with a Flat Earther. Using this simple knowledge it is also easy to just say something like this:

"Well we have known the Earth is a sphere for many centuries. Photos from space show a spherical Earth. Anyone surprised?" 

If you need more rigorous proofs of the things described here, then take a look here. 

1. Celestial Navigation (Eratosthenes)

Celestial Navigation is just a continuation of Eratosthenes' famous experiment from 230 BC detailed in the third experiment. Eratosthenes assumed the Earth was indeed spherical and with this assumption, he calculated the circumference of the Earth. This was simple maths: The difference in Sun angle between Alexandria and Syene was measured to 7 degrees. So from this, the circumference of the Earth can be calculated as D x (360/7) where D is the measured distance between Alexandria and Syene, which Eratosthenes estimated to be about 700 km. Using this he got an eerily accurate figure for the circumference of the Earth. 

However, Eratosthenes only measured using two measuring points, and the Earth being a sphere wasn't 100% proven during his time. It was a very strong conjecture, though (backed up by evidence from Aristotle and many others), but still not proven in the strict scientific way we require today.
It is not uncommon to hear Flat Earthers argue like this: "Eratosthenes' result would also work on a flat Earth". Well, they are right! You would see something similar even on a flat Earth with a local sun. But the story doesn't end at 230 BC.

If we continue and repeat Eratosthenes' experiment by adding more and more measuring points and observation locations, we will start to see something: The measurements always seem to fit into a spherical model. The altitude of any star (including the Sun) in degrees is always 90-(distance/111.1), where distance is measured in kilometers from the observer to the so-called Geographical Point (GP), i.e. the point where the star (or the Sun) is straight overhead at the time of observation. This is only possible on a spherical Earth with very distant celestial objects. Syene was Eratosthenes' GP. But we can make a catalog ("almanac") with known GP:s for stars at any time. And this has been done. The GP:s are listed in so-called Nautical Almanacs and we use sextants to measure the altitude. 

This is the only gear you need, together with some not-so-complicated maths: 

The mathematics is quite simple: Your task is to draw (at least) two circles on a spherical surface. The width of the circles shows the altitude (angle) of the observed star (or the Sun). If the star is almost above you, the circle will be very small. The lower the altitude, the wider the circle will get (90 degrees - altitude) x 111.1 km. The center point of each circle is the point on Earth where the star is in the zenith, and this information is picked from a nautical almanac. To pick the right information from the almanac, you use an accurate clock.

Your location is at one of the intersections.

If you make three or more measurements, you will see all circles intersect at (or near) a common point.

The only tricky part is to do the actual mathematical calculation of the intersection points.
This article describes this schematically.
But for good precision, you need to make small adjustments. Atmospheric refraction, the dip of the horizon, and the shape of the Earth (it is not perfectly spherical) require some small adjustments.
Professional navigators on the oceans typically use pre-calculated tables for this: So-called Sight Reduction tables. But you can also use calculators or computers. I have made a computer software toolkit, which you can read about here.

The simple fact that celestial navigation works consistently is enough evidence for a spherical Earth. It also proves that Eratosthenes' assumptions about Earth's sphericity and very distant celestial objects were right. For anyone questioning this, I have prepared the fourth (bonus) challenge: An impossible task to design celestial navigation on a non-spherical model (such as Flat Earth).

If you have time, I recommend checking this video. 

A more rigorous proof can be found here.

2. Al-Biruni's Experiment (dip of the Horizon)

The other simple experiment proving Earth's sphericity does not depend on anything outside our planet. You just have to look at the horizon, and if you look carefully (with a good level) you will see something interesting: The horizon dips, and the dip increases with your elevation. In other words: The horizon is not at eye level! Even from a very low elevation, 1-2 meters, you can measure the dip with precise instruments, such as a theodolite or a so-called total station.

Everyone doing this will see a fundamental thing: A square root relation between elevation and dip angle.
More precisely: The dip in arcminutes equals 1.85 times the square root of elevation in meters.
Let's check this:
If the elevation is 1 meter, you will get 1.85 arcminutes dip.
If the elevation is 100 meters, you will get 18.5 arcminutes dip. (The square root of 100 is 10)
If the elevation is 10000 meters, you will get 185 arcminutes dip. (The square root of 10000 is 100). 185 arcminutes is about three degrees (an arcminute is 1/60 of a degree). Three degrees is a large angle!
You can check this yourself at any time.
This relation is very accurate for lower elevations but needs to be replaced with a trigonometric formula for large distances from Earth.

The underlying maths for this is quite simple; you can read more here. Since this square root relation is visible in all locations and directions and follows the defined pattern, it is very strong evidence for a spherical Earth.

In the picture below, you see the dip. It is the angle between the true horizon and the astronomical horizon. The dip increases the higher you climb.

This experiment has been performed countless times by surveyors worldwide, always giving the same result. This video shows how this can be done using professional gear. 

However, the experiment is also available to laymen. 

1. Using a simple water level, you can at least see the dip clearly (but not measure it accurately). 

2. If you visit a skyscraper, you can try this: Stay on the bottom floor and see the sunset. As soon as the sun sets, take a rapid elevator to the top floor or call a friend who is already there. You will then see a second sunset (or your friend will see the sunset several minutes after you). This is easy to calculate. Assume you are at Burj Khalifa in Dubai. The top floor is at about 550 meters elevation. 1.75 times the square root of 550 is about 41. 41 arcminutes is wider than the angular diameter of the sun. So at the time when the sun sets at the bottom floor, it is visible entirely over the horizon at the top floor. You can call a friend up there, or take a rapid elevator. 

3. If you have access to a drone with a camera, you can try this experiment. 

4. The fourth and simplest experiment: From high elevation in an airplane, the horizon dip has increased to 3 degrees at 10000 meters. This requires no fancy gear at all. You only need a half-empty water bottle. 

The horizon dip was discovered by Persian astronomer Al-Biruni from a mountain in today's Pakistan in 1017 AD. 

A more rigorous proof can be found here.