# What is the simplest way to prove the Earth is round?

Assume you've come in contact with a tribe of people cut off from the rest of the world, or you've gone back in time several thousand years, or (more likely) you've got a numbskull cousin.

How would you prove that the Earth is, in fact, round?

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That depends on the definition of "round". – Leos Ondra Sep 17 '13 at 6:38
At least two thousand years see en.wikipedia.org/wiki/Spherical_Earth#History – WetSavannaAnimal aka Rod Vance Nov 19 '13 at 12:31
This question appears to be off-topic because it is most appropriate on earthscience.stackexchange.com – BMS Dec 24 '14 at 22:59

The shadow of the Earth on the Moon during an eclipse and the way masts of ships are visible when they are out of sight are the classical reasons.

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Simplest, you say? There are two that strike me as being simple to demonstrate. Luckily someone on the internet has already spent some time to help us here to make these easy to illustrate:

# 1. Shadows differ from place to place

Eratosthenes carried out this experiment to determine the circumference of the Earth, already assuming its spherical shape; incidentally, the proof of such being consequential of the procedure.

However, a demonstration can be achieved by a simple, local experiment (as opposed to having a party venture to a distant enough point):

Take a piece of card (A3, or so), attach two obelisks to the card by their bases and, with a light source, produce shadows - now, slowly bend the card so that it becomes convex (that is, the side with obelisks attached bulging out) and watch the effect.

# 2. You can see farther from higher

There are numerous other ways of demonstrating that the Earth is round, or curved, at least, from analysing the center of gravity to simply observing the other round objects that are visible in space; but I believe these illustrations to be the simplest to comprehend.

Images sourced from SmarterThanThat

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Another way is the triple-right triangle:

1. You move in a straight line for a long enough distance
2. Turn right 90° degrees, walk in that same direction for the same distance
3. Turn again to the right 90° degrees and walk again the same distance

After this you'll end up at the starting point. This is not possible on a flat surface since you'd just be "drawing" a half-finished square.

Source: http://www.math.cornell.edu (add /~mec/tripleright.jpg to find the image)

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"Walk"? At some point, you'll have to swim. :) Which might be a welcome change from a year of walking... – cHao Apr 4 '14 at 4:18
@cHao Or you can use a hot-air balloon. :D – Alenanno Apr 4 '14 at 9:28
What is the minimum distance for each side? – RedSirius Apr 14 '14 at 15:43
@RedSirius I asked that very question some time ago. :) – Alenanno Apr 14 '14 at 17:17
What's a practical way to ensure one is both turning through exactly 90 degrees and proceeding in "straight" lines? – Todd Wilcox Feb 2 at 18:50

Sitting for a while by the seashore ought to make it clear the Earth isn't flat, even if you don't happen to see a ship go over the horizon. The edge of the discworld Earth would have to be just a few miles away, and there's no way that the entire, circular world would fit inside the circle that the ocean horizon seems to make.

Humans have not just known the Earth was spherical but actually have been measuring its radius for thousands of years. http://en.m.wikipedia.org/wiki/History_of_geodesy

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You can build a simple pendulum and observe how it rotates as the day progresses. You can then put a pendulum on a stick or something that you can rotate yourself in order to demonstrate that when you rotate the stick, the pendulum will continue to swing in the same direction. This shows that the direction of movement of the pendulum will change relative to its base only if its base is rotated.

Pendulums can also be used to measure your latitude (its direction will change at different rates for different latitudes), and to measure the local value of g (the amount of time it takes to go through one cycle, or its period of oscillation, will vary with gravity).

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Doesn' t this merely prove the Earth rotates? – Andrew Jun 24 '11 at 10:21
@Andrew D'oh! I'm tired and assumed they meant the same. I guess by extension it proves the earth is round, assuming you're not doing this experiment at one of the poles. – Carson Myers Jun 24 '11 at 11:04
It's also surprisingly hard to build a decent Foucault pendulum. It is necessary to insure that the suspension is torque-free, which is not easy. – dmckee Feb 26 '12 at 22:45
@dmckee torsion-free suspension is only part of the problem. I have seen the experiments described here from close up. Sir Brian Pippard was arguably a very, very good experimental physicist... but he was stumped. Well - he made a great pendulum, just not great enough to look for confirmation of the Thirring-Lense effect – Floris Jan 27 at 20:49

If the person in question is from a temperate latitude, take them to the tropics to feel the heat of the noon sun, preferably trapped out on a sailboat without water. Point to the very high sun and make your point when they are the most miserable. Next, take them to a very high latitude. As they freeze and become exhausted at 3:00 AM while out walking the tundra, point out the low, non-setting (or non-rising) sun, and re-iterate your point in their heightened state of misery. Through suffering and a sense of pride, the object of your demonstration will now likely feel that they have "been there" and "seen it" with "their own eyes". If convinced, that person will gladly proselytize the "truth" of aforementioned roundness of said planet, and will confront the heretics who do not believe.

I think that there are no simple answers to provide "proof" of anything. "Proof" is relative, much in the way "truth" is relative. If simple means "without using science or technology" then you are without hope, as the receiver of the "proof" must accept the truth of the methodology.

Photos from space are photoshopped.

Ships at sea look below the horizon because Osirus/Neptune/Odin/Jesus/Bhaal does not wish man to see to infinity (which also proves that the heavenly bodies are not very far away).

Sticks in the dirt and shadows prove nothing unless you accept that other bodies are permanent, in orbital motion, and far away (at which point the person will already believe that the planet is round).

Don't try to prove anything. You can't. Instead, "Demonstrate and educate", because all you can do is convince, not prove.

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+1, the power of 'torture' ? ;) Fernão de Magalhães paid with is life to prove that the Earth is round. (in eng. is Ferdinand Magellan) – Helder Velez Jun 28 '11 at 2:01
Proof also connotes finality. Once something is proven, it need never be considered again. – user11266 Feb 20 '13 at 15:32

Besides the going back in time option, you could just show your "numbskull cousin" a picture of the Earth taken from the moon like the one below.

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Like the approach, side stepping an academic approach. Also you could show them picture/video from the ISS or this great video of James May taking a trip in a U2 spy plane where you can clearly make out that the Earth is round from 13 miles up: youtube.com/watch?v=x6cZLfK4Zjk – Fergal Jun 27 '11 at 11:22
I just find that sometimes when you are going for "simplest way", you have to tailor it to the audience. Some audience members aren't going to understand the math or the academic approach and a different way to show them is necessary. – Annika Peterson Jun 27 '11 at 16:23
What if your "numbskull cousin" doesn't believe in the moon landing? It's all fake! – jkeuhlen Dec 24 '14 at 20:53

I think the simplest way is to have two sticks of same size put both of them perpendicular to the surface of the earth in the mid day sunshine and the gape between them is to be few miles and exact time mesure the angle of elevation or mesure the size of the shadow so both will be differ! By several exams in a sysmetic order we can find that the earth is round.

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There's a video on youtube of a island a few miles away such that when you see the island from an elevation, you can see further to its base than you can when you see the island from the shoreline ( a demonstration of answer 1 above). I think this is the simplest way given that now we have zoom ability, anyone can do this kind of experiment on a clear day from any shoreline viewing something a few miles away.

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The occurrence of noon (i.e. meridian passage of true Sun) isn't simultaneous for two observers situated along an east-west line. Hmmm...okay perhaps even simpler. Sunrise and sunset aren't simultaneous for those two observers.

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Classically, the gravitational force experienced by a mass $m$ above the Earth is given by the familiar,

$$F=G\frac{Mm}{r^2}$$

where $M$ is the mass of the Earth. In other words, a mass will experience a force which continually decreases as it distances itself away from the Earth. Now suppose the Earth was a flat infinitely$^{\dagger}$ large plane in $\mathbb R^3$ which is infinitesimal, with mass density $\sigma$ (per unit area, not volume). The gravitational potential $\Phi$ satisfies the Poisson equation $\nabla^2 \Phi = 2\pi G \sigma \delta(z)$, assuming the plane is at $z=0$.

The solution is given by $\Phi(z)= 2\pi G \sigma |z|$. The gravitational force is $-\partial_z \Phi$, which is always pointing towards the plane. Another feature is that the gravitational force is constant with magnitude $2\pi G \sigma$. In other words, no matter how high one is above the plane, the same forced is experienced. To be more realistic, if the plane had some non-zero thickness, the force would still be constant, but whilst inside there would be a 'jump' as depicted:

Hence, to determine if the Earth is flat, one would simply have to conduct an experiment to see how the gravitational force scales as one increases altitude. One will find $F \sim r^{-2}$ approximately, as expected, confirming the Earth is round. Of course, for sufficiently small variations in $r$, one may be fooled into thinking $F$ is constant since the change is minute, but it is measurable.

$\dagger$ For convenience, it is taken to be infinitely large; the conclusions remain the same, but the force will of course be different, since it will be dependent on $x$ and $y$ as well.

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I like this answer because it is one the knowledge - impaired cousin can easily understand – WoJ Dec 22 '15 at 6:21

draw a triangle. On a curved surface the angle sum of a triangle is never equal to 180. If that is the case on Earth, it is spherical.

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Wait, I accidentally flagged this as not an answer. Sorry, @Moderators: please decline the flag. +1. Except that this is not the simplest way. – centralcharge Sep 23 '13 at 15:03
Have you analyzed how large a triangle you would have to draw to measure the deviation? Why the proposed method is correct this is in no way the "simplest" method due to the practical difficulty of the task. – dmckee Sep 23 '13 at 15:28
@dmckee, yes I agree – Shreesha Einstien Sep 24 '13 at 4:01
@DImension10AbhimanyuPS, Dude why are you always behind me. This is a physics discussion forum, lets not fight here. – Shreesha Einstien Sep 24 '13 at 4:02
+1 although your method detects curvature, it does not prove that Earth is spherical. If $\sum \alpha_j>\pi$ everywhere, then you can prove that the Earth's surface is a compact and closed manifold, which is getting close (but you still have surfaces of different genus). If there are places where $\sum \alpha_j<\pi$, then you can't say much aside from "it's curved". – WetSavannaAnimal aka Rod Vance Nov 19 '13 at 12:26

## protected by Qmechanic♦Nov 19 '13 at 11:02

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