# How one can establish that the Earth is round?

I am asking this question out of curiosity.

Currently, I am reading the book: A Brief History of Time. In Chapter 1, the author discusses about Aristotle's conclusions about spherical Earth.

For example: At horizon, one sees the top of the ship first and then its base. Also, the shadow casted by Earth on the Moon's surface is round during the lunar eclipse. Thus, one can conclude that the Earth is round.

In what other way, one can conclude that the Earth is round? Obviously, without observing it from a space station or using some modern technology. Or put in other way, what are some other evidences, that people in Before Common Era used to conclude that the Earth is indeed round?

• Related: Eratosthenes calculated the circumference of the Earth around 240 BC en.wikipedia.org/wiki/… Commented Jun 28, 2023 at 19:47
• Here is a famous clip from the documentary "behind the curve" where a group of flat-earthers conduct an experiment and accidentally prove that the earth is curved. Commented Jun 29, 2023 at 12:08
• There are similar questions on the physics site that you might be interested in. physics.stackexchange.com/questions/26427/… physics.stackexchange.com/questions/232114/… Commented Jun 29, 2023 at 13:59
• Do remember that for many proposed experiments, one should be clear on the preconditions that one takes for granted. E.g. what is the distance of the sun/stars from earth, and is it valid to consider their rays as (effectively) parallel to each other? Does the sun shine like a ball in all directions or like a flashlight in only a small cone? How far does visibility travel? Many comments point caveats out. Flat earthers come up with many possibilities that should be taken into account - it is never just about proving one thing but proving each "turtle" all the way down.
– frIT
Commented Jun 29, 2023 at 14:07
• Technically, the Earth is not round. Or a sphere. Heck, technically it's pear-shaped instead of an oblate spheroid. But still I say that the Earth is round... Commented Jun 29, 2023 at 20:17

Observer the shadow of the Earth on the moon. It is circular, no matter where the moon is in the sky.

Observe that the position of the stars in the sky changes as you travel north or south. In particular, the pole star is higher in the sky the further north you go. Similarly observe that on midsummer, a vertical stick in Syene casts no shadow, but in Alexandria a stick casts a shadow at 7 degrees.

Note that the sun and moon appear to be spherical and suppose the Earth to be similar.

Consider what shape a large amount of matter will form into under a gravitational force: that shape is a sphere.

Travel around it (okay nobody did this 2000 years ago, but some people (Pacific Islanders for example) could.)

Survey the angles in a large triangle. Observe that they add up to more than 180 degrees.

• @planetmaker, the OP specifically asked for evidence other than a ship disappearing below the horizon. Commented Jun 28, 2023 at 22:50
• How do you know that the shadow on the moon is caused by the Earth? :-) Commented Jun 29, 2023 at 11:22
• "Survey the angles in a large triangle. Observe that they add up to more than 180 degrees." - This appears to be a reference to the geodesic research of Carl Friedrich Gauss in the early 19th century. Commented Jun 29, 2023 at 11:58
• Syene/Alexandria: this would also hold true for a flat earth with a sun sufficiently close to the surface (i.e. the distance of the round earth's radius) and obviously pro rata smaller/dimmer. (Sagan's claims in the internet video notwithstanding, the Greeks already new the earth was round from the moon shadow, and only used this to calculate the circumference.) For (at least) 3 sticks/pillars/wells, you'd need sufficiently large distances/heights and accuracy to see the difference from the flat earth model - could probably be done though.
– frIT
Commented Jun 29, 2023 at 13:53
• @GlenYates Or you can trust and accept the observations of other people who waited for an eclipse and did it, rather than supposing that the entire scientific community is engaged in some massive conspiracy and falsifying exactly and only those observations which they know you aren't going to verify yourself. Commented Jun 30, 2023 at 8:34

### Aristotle's arguments

Aristotle used the following four arguments for a spherical Earth (of which James K already discussed three):

1. During a Lunar eclipse, Earth's shadow is always round. If Earth had any other than round, you would sometimes see another shape.

2. The position of the stars are clearly different when observed in the north and in the south. For instance, he noted that some stars seen from Egypt and Cypres weren't visible from Greece at all.

3. In ancient times, there were elephants in North Africa, at least near the Strait of Gibraltar. If Earth were round, this would explain the existence of elephants there and in India, but not in Europe: The could simply walk around the other side.

4. One of the counterarguments against a round Earth was that people on the other side would fall off. Aristotle made this argument not a problem, but a solution: Massive things have a tendency to fall toward the center of the spherical cosmos, and if particles come from all sides toward one point, the resulting mass must look the same from all sides, i.e. spherical.

Argument #3 is… well… not the best argument. Argument #1 could in principle — and is in fact today by modern flat-earthers — be explained by lunar eclipses not being caused by Earth, but by a mysterious invisible "shadow object". Argument #2 can also be explained by the stars being much closer to Earth than we think. Finally, argument #4 is an fascinating prediction of how gravity really works, but at Aristotle's time it was mostly speculation.

### Arguments that Aristotle did not use

Interestingly, despite being available to him there were several arguments that Aristotle didn't use:

One of the most common arguments for the sphericity of Earth's is that fact that you see the mast of an arriving ship before you see the hull. However, in certain ancient cosmological models, the Earth was not completely flat, but slightly curved like a shield, which could produce the same effect. Perhaps that is why Aristotle chose to disregard this argument.

Phenomena such as the sunset happen at different times, depending on where on Earth you are. Travelers from the north reported that the day was shorter up there, and the days passed faster when traveling from west to east than when traveling from east to west$$^\dagger$$. It is difficult to explain on a flat earth why the Sun cannot be seen by everyone at the same time, unless one simply assumes that the Sun shines like a kind of flashlight with a cone of light that only hits a limited area on the Earth's surface. This explanation was used, for example, in the flat-earth Gai tian cosmology of the 1st century BC. (but which is based on the roughly 1000-year-old world view Zhou Bi Suan Jing). It is also used by modern flat-earthers.

But the most important argument, which Aristotle did not make use of, is probably that when sailors crossed the equator, they saw the night sky revolve around a different star than what they were used to. To my knowledge this is not possible to explain on a flat Earth (but of course requires you to perform a journey, which isn't just something you can do while having a discussion with a flat-earther).

Today, we of course have many more ways of realizing that the Earth is round. However, as I described in an answer on physics.SE, none of these would enable you to convince a flat-earther since any argument that you come up with ultimately relies on your faith in previous people's observations and experiments (unless you want to derive all of physics and carry out all experiments in front of the flat-earther), and can thus be rejected either as "wrong" or "fake" or "how do you know that refraction works that way? Ah, you read it in a book?".

As others have noted here, the experiment carried out by Eratosthenes assumed, not proved, a round Earth.

$$\dagger$$In the encyclopedia Naturalis Historia from the 1st century AD the Roman historian Pliny the Elder describes how a messenger who ran the same route east and west noticed that the trip took longer from west to east, even though it was downhill.

• I don't follow the elephant argument, it's possible to walk from India to North Africa regardless of whether the world map is projected on a sphere or a flat plane. In reality, there is no way to "walk around the other side" - the premise that it's necessary is faulty in the first place, so the conclusion doesn't hold. Commented Jun 29, 2023 at 13:21
• @NuclearHoagie But there are no elephants in the territories between India and North Africa that were known to Aristotle, which one would expect if they had spread out to both locations from some point of origin. Unless they spread through territories unknown to Aristotle, i.e. the other way round a spherical Earth. The question isn’t how could elephants walk from India to North Africa, it’s why didn’t some of them stop halfway. Commented Jun 29, 2023 at 18:04
• @pela I meant that you can draw a walkable path from Africa to India on a flat map as well as a globe. The simple fact you can walk from Africa to India doesn't suggest the world a globe, and it wasn't clear to me why Aristotle expected the elephants would need a "back way" that could only be present on a spherical world. It seems he was working on the incorrect assumption that animal habitats are necessarily contiguous, which led to a correct conclusion (the earth is round) with an incorrect detail (that elephants could walk east from India to Africa). Commented Jun 29, 2023 at 20:48
• @NuclearHoagie Ah okay, I misunderstood. Yes, elephants could have migrated like humans do (and of course did), but I suppose his argument was that you would then expect to also find elephants in the rest of North Africa and in the Middle East. Moreover, in Aristotle's "Meteorologica" he actually says that "beyond India and the Pillars of Heracles [i.e. the Strait of Gibraltar] it is the ocean which severs the habitable land and prevents it forming a continuous belt round the globe", which contradicts his own argument.
– pela
Commented Jun 30, 2023 at 8:39
• @IlmariKaronen I think the Greeks didn't sail so far south, but there had been reports of Phoenicians mariners going round Africa at the time of Parmenides (i.e. ~100 years before Aristotle), and Herodotus wrote in his "Histories" from 430 BC that "One of their claims — which I personally find incredible, although others may not — was that, while sailing round Libya [i.e. Africa] they had the sun on their right hand side [meaning in the northern sky]" (I haven't read "Histories" myself, but found this in the book "When the Earth was flat" by Dirk Couprie from 2018).
– pela
Commented Jun 30, 2023 at 8:55

Here's my favorite method. It's easy, can be performed most places on Earth, and doesn't require any calculations at all, merely simple logic.

Take a transparent vessel, preferably one that does not diffract light that passes through. The easiest one is likely a plastic water bottle you can get at any gas station or convenience store. (If you can't get ahold of a suitable transparent vessel, any vessel with a flat top that has been filled to the brim will suffice.)

Get to a place where you have an unbroken view of the horizon. Higher elevations such as a mountain or tall hill are preferred as they will produce a more pronounced effect.

Hold the water bottle close to your face so that the surface of the water in the bottle is level with your eyes. This will in effect guarantee that you are looking "straight forward" relative to gravity's "straight down".

Look at the horizon and note where it lies relative to the water level. For a flat Earth, you would expect it to align with "straight ahead" as the surface would stretch out ahead forever (or at least far enough to effectively be forever). Instead, you will see that it is slightly below where you are looking, which is what you would expect if the Earth was curving away from you.

(Source)

Eratosthenes of Cyrene, a chief librarian of Alexandria, is cited as the first person to determine the circumference of the Earth in about 300 BCE. By knowing the distance between two locations on the same north-south meridian and measuring the angles made by shadows at the same time on the same day in each one can do a geometrical calculation to determine the circumference.

The Egyptian city of Syene was situated nearly on the Tropic of Cancer, so on the summer solstice the sun passed directly overhead, casting no shadow at high noon. Eratosthenes and/or assistants made the measurement of shadows at noon on the solstice in Alexandria. Thanks to extensive surveying in the Nile valley and records at the Library of Alexandra, the distance between the cities was well known. The result of $$252\, 000$$ stadia is a few percent off from the modern measurement mostly due to uncertainty in the conversion between stadia and modern units, like meters.

• Assuming a flat earth, you could use the same measurements to determine the altitude of the sun. For this to work you had to already assume the earth is round. Commented Jun 28, 2023 at 22:04
• @antlersoft Of course, adding in a third city (say, Istanbul) to your observation allows one to distinguish between the flat and curved models, as the flat case would yield inconsistent sun heights $h = d/\tan(\theta)$ (where $d$ is distance to Syene and $\theta$ is the noon solstice shadow angle). If $d = R \theta$ indeed holds as on a spherical Earth, the inferred sun height will vary with $\theta$. Commented Jun 29, 2023 at 15:06
• This Stand-up Maths video shows two people recreating this. Commented Jul 1, 2023 at 19:39

The angle that Polaris is roughly equal to the latitude. and difference in latitude is roughly proportional to distance travelled in the North-South direction. On a flat plane, the angle to Polaris would also change with latitude, but the change wouldn't be constant.

While Eratosthenes' measurement of the Earth assumed a round Earth rather than proved it, doing that experiment at different latitudes would, on a flat Earth, give different answers, so different people getting consistent answers shows that the Earth is round.

The stars rotate clockwise around the North pole in the Northern hemisphere, but counterclockwise around the South pole in the Southern hemisphere. Polaris isn't visible in the Southern hemisphere, and the Southern cross isn't visible at Northern latitudes.

The Northern and Southern hemispheres are symmetric. In most flat-Earth models, the Northern and Southern hemispheres are very different.

The area of a circle is smaller than $$\pi r^2$$, and the circumference is less than $$2 \pi r$$.

Above the Arctic Circle, the Sun doesn't set on the Summer solstice.

Also, night exists.

People talk about seeing a ship come up over the horizon, but really the very existence of a horizon is a clue. When you stand on the shore and look out at the ocean you see a sharp line, as if the ocean were a giant infinity pool, but you know that's not the edge of the earth because you can go that far on a boat. Even on land, everywhere you go that's kind of flat there's a horizon that things can come up over, rather than just fading in out of the haze.

So that means the earth is curved. And you don't really have to go that far to go over the horizon. So it seems like the curvature must be pretty significant over thousands of miles. That's where things like ship's masts are helpful because you can triangulate with them and get an actual estimate.

And here, I think, is an interesting point. Once you know that the earth is curved at all, even if you don't have further evidence that it's actually a ball, that starts to look like a pretty attractive hypothesis. Naively, if the earth were a big dome, all the water should run off it. As soon as you convince yourself that the real desire of water is to run in the "locally down" direction, which ends up being towards some point at the center of curvature of the dome, then why not just assume a whole ball?

I want to be clear - that, by itself, doesn't prove it's a ball. It was the earth's shadow on the moon, plus other things like the angle of the stars in the sky, that really convinced the Greeks. But you always interpret evidence as a totality. It's a lot easier to accept the eclipse evidence once you're convinced that a curved earth isn't impossible.

• I agree, but I think the big problem is that you can't really see what's going on at the horizon, especially with the naked eye, but also with binoculars. Mist and refraction and the mere distance tends to obscure what actually is at the horizon. So a flat-earther would say "Yes, you can't see to the other end of the world because it's too far." Moreover, modern FE'ers have some wac… alternative explanation of how sightlines don't extend to infinity, but terminate on the ground many kilometers away (I discussed this with a FE'er on a radio show once). Anyway, +1.
– pela
Commented Jul 2, 2023 at 20:11

If you were flying in an airplane and the Earth was flat, you would be able to see all the cities of the world with magnification, but you can't even see a city that is a 3-hour flight away because it's beyond the horizon of a round Earth.

• That wasn't really available until the invention of airplanes, so not in the time of the ancients. Commented Jun 29, 2023 at 23:44
• On a flat Earth, you can't see a city that's a three-hour flight away because it gets lost in the atmospheric haze. The distance limit for visibility in even a perfectly clear atmosphere is about 300 km.
– Mark
Commented Jun 30, 2023 at 2:43
• Whoops, I missed the "Before Common Era" part. Just swap out "flying in an airplane" for "standing on top of the local cathedral". Commented Jun 30, 2023 at 13:41

One thing not mentioned yet (except for the ship/mast phenomenon) is the curvature of a large body of water (e.g. reservoir, dam or lake). This rests on the assumptions that

1. the body of water as a small section of the whole earth is representative of the whole earth (physics work the same everywhere and don't have local quirks)
2. the earth is in fact (approximately!) spherical (and not shaped like a lens, Capt America's shield, etc.).

I'm too lazy to do the trigonometry myself, but using the online calculator at https://planetcalc.com/1421/, I plug in the following values:

The results then show me:

• I need a body of water that has a stretch of about 5.6 kilometers between two points on its shores (how that is determined accurately and whether you can trust the online/paper maps is left as an exercise to the reader).
• One should then have a "bulge" of water of about 0.61 meters between the two points.
• So if one would put an appropriate telescope on the ground at the first point (at water level) and a vertical stick with markings at the second point, one should only see the markings that are higher than 61 cm on the stick looking through the telescope, the lower ones being obscured by water. Provided that visibility over water between the to points is clear *, and waves are to a minimum!

(* = It has been pointed out on the internet that the visibility near the surface of water is often distorted due to the temperature difference between the water and the air, so ideally one would want to have 3 reference points raised the same distance off the surface of the water at the two points and at the midpoint - which complicates the experiment considerably.)

After writing this up I also did a websearch on this problem and found this article on the power transmission lines on Lake Pontchartrain (one of many). The process is described, but it also supplies multiple photos of a convenient existing structure.