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Prove that the Earth is a sphere

We know that planets are spherical, not flat.

Gravity causes matter to pull inwards, eventually into a spherical shape. This happens for stars, planets, moons and even some asteroids.

Nontheless, some people do not believe that the world is spherical. Flat Earthers exist in larger numbers than one might imagine, and they literally believe that the Earth is flat. How do they maintain this belief, despite all proof? They are engaged in a thought process called conspiracism

Conspiracism is neither a healthy expression of skepticism nor a valid form of criticism; rather it is a belief system that refuses to obey the rules of logic. These theories operate from a pre-existing premise of a conspiracy based upon careless collection of facts and flawed assumptions. What constitutes “proof” for a conspiracist is often more accurately described as circumstance, rumor, and hearsay; and the allegations often use the tools of fear—dualism,demonization, scapegoating, and aggressively apocalyptic stories—which all too often are commandeered by demagogues.

Chip Berlet, Toxic to Democracy: Conspiracy Theories, Demonizaiton and Scapegoating

Conspiracism: Center for Academic Success, Butte College

In the ancient Near-East: Babylon, Judea/Israel, Syria, Mesopotamia. In these cultures, the world was portrayed as a flat disk floating in the ocean. It was speculated that there were waters in the Heavens above, and waters gathered in the deep below. This idea is found in most religious texts from the ancient near-east. Some statements about this exist in the Hebrew Bible (Old Testament.)

As far back as the Greek philosophers Pythagoras (6th century BCE) and Parmenides (5th century BCE) it was recognized that the Earth is spherical.

During the Medieval Era Aristotelian physics and Ptolemy’s astronomy was accepted by most Christians, Jews and Muslims. (The Medieval period covers roughly 5th to the 15th century, from the collapse of the Western Roman Empire up to the Renaissance.)

How can we show that our world is spherical?

If the Earth is a flat disc then there is no south pole, only a north pole see image above.

So as this 2D, flat Earth rotates, we’d see all stars circling around Polaris, the “pole star.” People would see that from anywhere on this supposed flat Earth.

In contast, on a 3D spherical Earth, there’d be two poles; from the north pole we’d see the pole star, and northern hemisphere stars seeming to circle around it.

Yet from the south pole, Polaris wouldn’t be visible. Instead, we’d be looking at the opposite side of the sky – stars on the other side of our galaxy (imagine the galaxy sliced through it, like a sandwich)  We’d see these other stars seeming to circle around the point over the southern pole.

This has been tested countless times. Visitors to the poles have taken photos. What did these see?

  • On a flat, circular, 2d world: both photos would show same stars in the sky, with the north star being focal point.

  • On a spherical world: one photo shows the north star, and neighboring stars, and other shows the opposites. And that indeed is what we see.


This section is based on the article by Moriel Schottlender from PopSci.com, but with editing, and additional diagrams & animations added by Robert Kaiser

1. The Moon

Aristotle noticed that during lunar eclipses (when the Earth’s orbit places it directly between the Sun and the Moon, creating a shadow in the process), the shadow on the Moon’s surface is round. This shadow is the Earth’s, and it’s a great clue on the spherical shape of the Earth.

For example, you can see Earth’s round shadow on the moon during this total lunar eclipse, visible from Indianapolis, Feb 2008. ©bigstockphoto.com/alexeys


If the Earth was flat, then occasionally people would have seen Earth’s shadow, like on this (obviously fantastic) image.


Since the earth is rotating (see the Foucault Pendulum for definite proof), the consistent oval-shadow Earth produces in each and every lunar eclipse proves that Earth is not only round but spherical.


Foucault pendulum GIF

2. Ships and the Horizon

Approaching ships do not just “appear” out of the horizon – like they should have, if the world was flat – but rather emerge from beneath the sea.

Ship hull disappers horizon.jpg

Here is an actual photo of ships coming towards Galveston, Texas.

Note that the ships further away are “hull down”; their hulls are hidden by the curvature of the Earth.

3. Varying Star Constellations

This observation was originally made by Aristotle (384-322 BCE), who declared the Earth was round judging from the different constellations one sees while moving away from the equator.

After returning from a trip to Egypt, Aristotle noted that “there are stars seen in Egypt and…Cyprus which are not seen in the northerly regions.” This phenomenon can only be explained if humans were viewing the stars from a round surface.

Aristotle continued and claimed that the sphere of the Earth is “of no great size, for otherwise the effect of so slight a change of place would not be quickly apparent.” (De caelo, 298a2-10)

The farther you go from the equator, the farther the ‘known’ constellations go towards the horizon, and are replaced by different stars. This would not have happened if the world was flat.

Click this image to enlarge: the sky in the southern hemisphere looks completely different from the sky in the northern hemisphere.

4. Shadows and Sticks

If you stick a stick in the (sticky) ground, it will produce a shadow.

The shadow moves as time passes (which is the principle for ancient shadow clocks).

If the world had been flat, then two sticks in different locations would produce the same shadow:


But they don’t. This is because the earth is round, and not flat:


Eratosthenes (276-194 BCE) used this principle to calculate the circumference of the Earth quite accurately.

5. Seeing Farther from Higher

Standing in a flat plateau, you look ahead of you towards the horizon. You strain your eyes, then take out your favorite binoculars and stare through them, as far as your eyes (with the help of the binocular lenses) can see.

Then, you climb up the closest tree – the higher the better, just be careful not to drop those binoculars and break their lenses. You then look again, strain your eyes, stare through the binoculars out to the horizon.

The higher up you are the farther you will see. Usually, we tend to relate this to Earthly obstacles, like the fact we have houses or other trees obstructing our vision on the ground, and climbing upwards we have a clear view, but that’s not the true reason. Even if you would have a completely clear plateau with no obstacles between you and the horizon, you would see much farther from greater height than you would on the ground.

This phenomenon is caused by the curvature of the Earth as well, and would not happen if the Earth was flat:


6.  Ride a Plane

Planes can travel in a relatively straight line a very long time and not fall off any edges. They can also circle the Earth without stopping. If you look out the window on a trans-Atlantic flight, you can, most of the times, see the curvature of the earth in the horizon.


E. O Stinson addresses the math behind this issue on Quora:

You can absolutely see the earth’s curvature if you’re at altitude, particularly over the ocean in a plane. At sea level, you can only see about 3 miles. The average angle of human vision is somewhere between 120º and 140º; more if you count the whole image your brain builds up as your eyes move.

Which is a teeny number. According to Wikipedia, at 33,000 feet, the curvature is a whopping 5.6∗10 −2  “the same curvature of the rim of circle with a radius of 10 m that is viewed from 56 cm”.


Here is a research article on the topic

In view of the agreement between the visual observations, measurements of the photographs, and the theoretical curvatures, it seems well established that the curvature of the Earth is reasonably well understood and can be measured from photographs. The threshold elevation for detecting curvature would seem to be somewhat less than 35,000 ft (10.6 km) but not as low as 14,000 ft (4.2 km). Photographically, curvature may be measurable as low as 20,000 ft (6 km).

Visually discerning the curvature of the Earth, David K. Lynch. PDF paper

7. Look at Other Planets

There are certain characteristics all planets have, and it will be quite logical to assume that if all planets behave a certain way, or show certain characteristics – specifically if those planets are in different places or were created under different circumstances – our planet is the same.

In other words: If so many planets that were created in different locations and under different circumstances show the same property, it’s likely that our own planet has the same property as well. All of our observations show planets are spherical (and since we know how they’re created, it’s also obvious why they are taking this shape). Unless we have a very good reason to think otherwise (which we don’t), our planet is very likely the same.

In 1610, Galileo Galilei observed the moons of Jupiter rotating around it. He described them as small planets orbiting a larger planet – a description (and observation) that was very difficult for the church to accept as it challenged a geocentric model where everything was supposed to revolve around the Earth.

Jupiter's Galilean moons

8. She Existence of Timezones

The time in New York, at the moment these words are written, is 12:00pm. The sun is in the middle of the sky (though it’s hard to see with the current cloud coverage). In Beijing, it’s 12:00am, midnight, and the sun is nowhere to be found. In Adelaide, Australia, it is 1:30am. More than 13 hours ahead. There, the sunset is long gone – so much so, that it’s soon going to rise up again in the beginning of a new day. This can only be explained if the world is round, and rotating around its own axis. At a certain point when the sun is shining on one part of the Earth, the opposite side is dark, and vise versa. That allows for time differences and timezones, specifically ones that are larger than 12 hours.

9. The Center of Gravity
There’s an interesting fact about mass: it attracts things to it. The force of attraction (gravity) between two objects depends on their mass and the distance between them. Simply said, gravity will pull toward the center of mass of the objects. To find the center of mass, you have to examine the object.

Consider a sphere. Since a sphere has a consistent shape, no matter where on it you stand, you have exactly the same amount of sphere under you. (Imagine an ant walking around on a crystal ball. From the insect’s point of view, the only indication of movement would be the fact the ant is moving its feet. The shape of the surface would not change at all.) A sphere’s center of mass is in the center of the sphere, which means gravity will pull anything on the surface toward the center of the sphere (straight down) no matter where it’s located.


Consider a flat plane. The center of mass of a flat plane is in its center (more or less – if you want to be more accurate, feel free to do the entire integration process), so the force of gravity will pull anything on the surface toward the middle of the plane. That means that if you stand on the edge of the plane, gravity will be pulling you toward the middle, not straight down like you usually experience.


10. Images from Space
In the past 60 years of space exploration, we’ve launched satellites, probes, and people to space. Some of them got back, some of them still float through the solar system (and almost beyond it) and transmit amazing images over to our receivers on Earth. And in all of the photos, the Earth is (wait for it) spherical.

Deep Space Climate Observatory DSCOVR transit Earth Dark side of moon

A NASA camera aboard the Deep Space Climate Observatory (DSCOVR) satellite captured a unique view of the moon as it moved in front of the sunlit side of Earth last month. The series of test images shows the fully illuminated “dark side” of the moon that is never visible from Earth. The images were captured by NASA’s Earth Polychromatic Imaging Camera (EPIC), a four megapixel CCD camera and telescope on the DSCOVR satellite orbiting 1 million miles from Earth. From its position between the sun and Earth, DSCOVR conducts its primary mission of real-time solar wind monitoring for the National Oceanic and Atmospheric Administration (NOAA). https://www.nasa.gov/feature/goddard/from-a-million-miles-away-nasa-camera-shows-moon-crossing-face-of-earth


11. Use Trigonometry! (from Wikipedia)

The previous ten ideas were from Moriel Schottlender from PopSci.com

Abū Rayḥān Al-Bīrūnī  (973 – 1048 CE), known as Al-Biruni (Arabic: البيروني‎‎) was a Khwarezmian Iranian Muslim scholar and polymath. He is regarded as one of the greatest scholars of the medieval era and was well versed in physics, mathematics, astronomy, and natural sciences, and also distinguished himself as a historian, chronologist and linguist. He spent a large part of his life in Afghanistan. He is regarded as the “father of geodesy” for his important contributions to that field.

Abu Rayhan al-Biruni accurately determined the Earth radius by formulating a trigonometric equation relating the dip angle (between the true horizon and astronomical horizon) observed from the top of a mountain to the height of that mountain.



12. The Coriolis effect

This is a physical effect which we see in the real world, which is absolutely impossible on a flat surface. It can only occur on a three-dimensional sphere. For our unit on the Coriolis effect see Coriolis effect on Kaiserscience.

13. Photograph the horizon, and think …

Tony Miller writes

If you take great care to photograph the horizon right through the center of your lens with your camera very carefully leveled and at high resolution with a very high quality rectilinear lens you can actually view the slight curve of the horizon from much lower altitudes (even just a few hundred meters).

From just 200 meters elevation, on a 4000 pixel wide image at 94.4 degree field of view you should expect about 7 vertical pixels of rise out of the horizon circle (as shown in my calculator link). You can make this slight ‘bump’ more visible by compressing the width of the image to about 10% of the original width and stretching it vertically by a factor of 2–4x.

For example, if we do this to the image here (which also required slight rotation):


The results are (try it yourself!)


And since the horizon is below lens center we’re not seeing lens distortion, which would flatten out the horizon.

The rest is my commentary and thoughts on visual photographic analysis…

It would be difficult to tell, in general, from a simple photograph if what you are seeing is actual curvature or lens barrel (aka curvilinear distortion). You have to know how it was shot and cropped and what the FOV is.

If you have an uncropped image and the horizon goes through the exact center, then you can judge that photography as likely fairly accurate.


14. Flat Earth conspiracy theory debunked

People have seen the Chicago skyline -from across the other side of Lake Michigan. According to conspiracy theorists called ‘flat earthers’, this is literally impossible if the Earth is curved, therefore if someone sees this it must be proof that the Earth is flat. Unfortunately for the conspiracy theorists, (a) their math about the curvature of Earth is wrong, and (b) they are curiously omitting the fact that air can bend light, producing images that appear in a place that one might not expect. (See our unit on optics)




Click on this image to embiggen: it clearly shows that even when Chicago is visible from across the Great Lakes, most of the city is still under the horizon.

Learning Standards

2016 Massachusetts Science and Technology/Engineering Curriculum Framework

8.MS-ESS1-2. Explain the role of gravity in ocean tides, the orbital motions of planets, their moons, and asteroids in the solar system.

Next Generation Science Standards: Science & Engineering Practices
● Ask questions that arise from careful observation of phenomena, or unexpected results, to clarify and/or seek additional information.
● Ask questions that arise from examining models or a theory, to clarify and/or seek additional information and relationships.

A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas (2012)
…Gravitational interactions are non-negligible, however, when very massive objects are involved. Thus the gravitational force due to Earth, acting on an object near Earth’s surface, pulls that object toward the planet’s center. Newton’s law of universal gravitation provides the mathematical model to describe and predict the effects of gravitational forces between distant objects. These long-range gravitational interactions govern the evolution and maintenance of large-scale structures in the universe (e.g., the solar system, galaxies) and the patterns of motion within them….

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