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Not sure if this still falls under worldbuilding. Let's assume a civilization living with the following restrictions:

  • No access to the surface or ability to observe it (no sky, no atmospheric samples from the surface, no distant large objects, etc)
  • No means to see distant objects in a straight line (caves and tunnels will limit visible range)*
  • Almost all light sources are artificial
  • No knowledge about how deep underneath the surface they are (unless this can be deduced by the available information below).
  • No electricity or computers
  • No magical means to gather information

*technically, they could try and build these things through heavy terraforming.

But here are things they can check or do know:

  • Advanced knowledge (whatever is needed; math, physics, chemistry, etc)
  • The gravitational constant
  • Temperature, pressure, atmosphere
  • Density of surrounding rock and material
  • Natural features like underground rivers, lakes, lava, and so on.
  • Limited ability to change elevation through cave systems (1-2km perhaps? How much would be necessary, if this helps at all?)

Essentially, it boils down to this: Can you deduce the shape & size of the planet you are on, by logic and with the gravitational constant while locked in a confined space with no outside view?

Bonus: Do they need to know that a "surface" exists (that they are underground), to deduce this? If they don't know about any surface (or that concept), would they deduce it's existence during this process?

The key question is, if they can; not necessarily how. But that would be really interesting to read.

Edit: Any connections to the surface exist as needed to allow for the environment to exist (air, water), but if they do they are just unreachable and not known of. Even if they would happen to follow a stream of water, it would ultimately just come from a hole in some rock wall. I'm primarily interested in deducing information in such a limited framework, not in how to break the rules and change the framework itself (for example, by digging up to the surface), or how the framework can exist.

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  • $\begingroup$ Do they have wells opening in the surface? $\endgroup$
    – L.Dutch
    Oct 8 '21 at 10:40
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    $\begingroup$ Where do the gasses they breath come from? $\endgroup$ Oct 8 '21 at 10:44
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    $\begingroup$ Why are people assuming they must have air from the surface? We could easily imagine subterranean organisms that make oxygen (or whatever else people need) through chemosynthesis. $\endgroup$
    – EdvinW
    Oct 8 '21 at 11:01
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    $\begingroup$ I voted that way because the OP's description of them knowing about g, yet not knowing about where the air comes from made clear that their understanding of science is limited in certain areas. "Could" is dependant on their sciences and so, remaining moot ie. unspecified by the OP, I must vote opinion-based as frankly it's anyone's guess if they can figure it out or not - given we don't fully understand the limits of their sciences and understanding. As the OP asked me in a comment "How would you figure this out?" - which isn't what the question asked and would be off-topic anyhow. @KeizerHarm $\endgroup$ Oct 8 '21 at 13:17
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    $\begingroup$ @Spencer As someone below already pointed out, even if you dig a tunnel you still only have limited room to store all that dug-out rock. So, technically, they can dig tunnels, yes - but the options are limited (it's basically just reshaping the available area). Also, for the sake of this scenario, they can't just dig to the surface (for whatever reason; perhaps they are under an ocean, etc). $\endgroup$
    – Katai
    Oct 10 '21 at 18:54

10 Answers 10

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Trivially

(It's not a complex problem, and they have access to sufficient information)

but not easily

(they need to do some serious engineering projects to get the data.)

Steps

  1. conceptually figure out they are on a spherical planet, or at least know that this configuration is one possibility.
  2. Dig a horizontal tunnel. Horizontal as measured by plumb bob and/or spirit level. Observe the curvature of the tunnel relative to a light in the distance.
    Do the same with a tunnel at right angles to this first, to determine that the curvature forms a sphere, not a cylinder.
    This and some simple math gives them the exact curvature of the gravity over the length of the tunnel, thus gives them an exact distance to the core of their planet.

At this point they know they are on a spherical planet, and exactly how big is it up to their tunnel's level.

  1. Build a large tunnel network at this same level. Specifically, dig the tunnels at various angles such that they have a grid of location, at the same level, scattered over a geographically large part of the planet. At least several hundred km, if their planet is Earth-sized. We need a number of locations at the same gravitational altitude, but with various latitude/longitude combinations.
  2. Install a Foucalt's Pendulum at many locations in the tunnel network. The closer the pendulum is to the equator of the revolving planet, the slower it will rotate its apparent axis. On the equator it will just swing as released, not turning. At the pole it will rotate exactly once per day. If the tunnel network does not cross the equator of pole one can still get the exact latitude, but you have to beat your data with a very thick maths tome to get it to confess the answer.

At this time they know how large their planet is, from core to their tunnel.
They know its shape.
They know that it is rotating, and they know exactly how long a 360 degree rotation takes.
They know exactly where they are located on the planet, both in latitude (calculated) and longitude(relative to some arbitrarily selected line)

I know of no way for them to determine how deep they are. They might get some indication from echosounding, but a soil surface sucks at returning a sound wave that hits it from below. You are much more likely to detect rock type boundaries, water table, metamorphic transitions, etc...
Maybe they could use the increase in temperature as you go down, coupled with the concept of absolute zero temperature, to put an upper limit to the distance from their tunnels to the surface?

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – L.Dutch
    Oct 11 '21 at 17:58
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As an underground civilization, we can assume that they have an elaborate system of seismographs. It helps, after all, to know when the ceiling is about to come down on your head.

With knowledge about densitities of material and knowledge of physics, calculating the speed by which an Earthquake propagates through rock ought to be feasible.

Armed with this knowledge, and with a sufficiently sensitive seismograph, they can gather a lot of useful info. For example, they can detect how deep they are because Earthquakes reflect of the surface. And, if they have really good seismographs, they might be able to detect the echo as the quake goes round the world, allowing them to figure out both it's size and that is roughly spherical (as the echo appears to converge onto the epicenter, instead of bouncing of a flat surface).

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    $\begingroup$ +1 I think this is probably the only proposal which could feasibly determine the size of the planet from the inside, enough sensitive seismometers could give all sorts of information about the shape and makeup of their planet. Some more information and a beautiful animation is available here $\endgroup$
    – Kai
    Oct 9 '21 at 0:34
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    $\begingroup$ Even without echoing around the world, as they locate the centers of successive earthquakes they are soon going to notice that the earthquake zone is bounded top and bottom by regions that do not have earthquakes. As they analyze the boundaries they will soon see they are curved and form concentric spheroids. Maybe even plate tectonics. Since density increases downward some enterprising rebellious upstart is going to propose a nether region lies above where density drops so low that there is no rock. He'll be thrown into magma for his heresay, but one day they name a starship after him. $\endgroup$
    – Doug
    Oct 10 '21 at 4:37
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What they can do is start digging a tunnel in a straight direction, and shine a powerful light through it. How can they check that they are going straight? The always dig orthogonally to the direction defined by a line with a weight.

At a certain point the tunnel, following the curvature of the planet, will deviate from the straight line path of the light.

If they assume$^*$ the planet is a sphere, by measuring the distance at which this happens and with some basic trigonometry can deduce the radius of the planet.

$d =\sqrt{h(2R+h)}$

enter image description here

Why would they do this is another story. Maybe they want to build a fast road, and a straight path seems to be the most efficient way, and a certain moment they notice they can't see all the light at the end of it.

$^*$ The assumption can be done after noticing that the phenomenon happens in every direction they pick for digging the tunnel.

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    $\begingroup$ Your solution work but they will only find the radius beneath their feets. $\endgroup$ Oct 8 '21 at 11:19
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    $\begingroup$ I came here to give essentially this answer, if it wasn't already up. +1 $\endgroup$
    – Zeiss Ikon
    Oct 8 '21 at 11:31
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    $\begingroup$ This doesn't answer the question. $\endgroup$
    – D.J. Klomp
    Oct 8 '21 at 15:26
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    $\begingroup$ variation in air refractive index due to temperature, pressure, humidity, etc. can induce curvature in beams of light on the same order of magnitude as the curvature of the earth. If these aren't taken into account, they might conclude the curvature is the opposite that it is. $\endgroup$
    – Rick
    Oct 9 '21 at 11:22
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    $\begingroup$ I was going to suggest this, which is basically a subterranean version of the Bedford Level experiment, which checked if a straight 6mile (9.7km) section of canal was curved or not. Refraction and wind were a major issue; if the experiment is done on a still morning the correct result will be obtained. Underground, there is no local heating to get in the way, and draughts can be blocked $\endgroup$
    – CSM
    Oct 11 '21 at 13:50
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No, but...

Thanks to the Foucault pendulum, an invention from the eponym French genius (XIX century), your civilization can follow this reasonning :

  1. The planet is spherical and it turns on itself around an axis.
  2. They can find the poles by searching the point where the angular velocity is maximum. They can also find the equator (where the angular velocity is null). By the way they can also find the pole with a compass.
  3. With some trigonometry they can calculate the radius of the sphere beneath their feet.

However it is not possible to determine the radius of the planet which includes the part above them.

You can achieve the same result with a compass but it is less practical as it you need to have straight lines along very long axis following a lattitude parrallel.

Interresting fact : If they find a Pole they can determine the length of a day (the duration of one revolution of the pendulum)

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  • $\begingroup$ Very interesting, I didn't even think about the radius part - but now that you mention it, it makes sense :) $\endgroup$
    – Katai
    Oct 8 '21 at 13:38
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    $\begingroup$ How would you determine the planet is spherical instead of a rotating cube? $\endgroup$
    – D.J. Klomp
    Oct 8 '21 at 15:55
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    $\begingroup$ The compass will lead them to the magnetic pole. On their planet, the magnetic pole can be on the equator. $\endgroup$
    – user132647
    Oct 8 '21 at 19:12
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Limits

The theoretical maximum amount of rock that can be over a natural cave is 3,000 m (per https://en.wikipedia.org/wiki/Cave which gives no obvious cite). This matches well to the crushing pressure of rocks, though. Beyond that, even the hardest stone will be crushed down by the rock above it, to fill the void.

No caves have been found this deep. The deepest oil wells are deeper, though, up to 9.5 km. The deepest hole ever drilled was about 12 km, and the temperature at the bottom was above boiling point, so would not have supported life.

The deepest mine, and the deepest a human has ever gone, is 3.9 km.

On a planet with lower gravity, or in a species which lives underwater (so the pressure between rock and water is equalized and the caves don't crush so easily), far deeper caves would be possible.

So it seems there are answers already aplenty to establish what's below, and their position relative to it:

  • how fast their rocky universe rotates about an axis (Faucalt pendulum).
  • what angle they are to the axis (pendulum rotation = universe rotation rate x sine of latitude)
  • how far they are from the gravitational core (curvature of level tunnels, divergence of parallel vertical tunnels, etc)

Finding the distance to the surface is arguably a trickier proposition, but there have been many ways proposed for indirect measurements.

  • The pull of the mass above you. But inside a sphere, there is no net "pull of the mass above you", so gravitational measurements of the rock above you are not going to help.
  • Echo-sounding/seismographic surveys. On most planetary-sized objects, this would definitely work no matter how deep they are, at least with seismic/nuclear-level waves. Nukes are unlikely to be popular as scientific measurement tools for civilizations living underground, so only seismic effects would give a big enough wave to measure as it reflected around the world.
  • Rock type boundaries/Metamorphic transitions. This one is interesting, it requires understanding of rock formation and subduction, but could work as way to double-check other results.
  • Water table. Requires them to have a water table, mind, and I'm not sure this is a reliable estimate of depth from surface.
  • Geothermal gradients.
  • Air pressure, to get atmospheric height.
  • Digging upwards.
  • Liquid pressure measurement in a chamber. This would work, but is unnecessary: an important measure when mining is the pressure of the rock, and for this you can measure the deformation of a drilled hole using common mining equipment. An underground civilization would likely have improved over our own approaches for this. https://geoinfo.nmt.edu/publications/monographs/circulars/downloads/69/Circular-69.pdf explains the basics (at least as they were in 1963: I imagine we use much more sensitive electronic measures now)

Can we make a planet which makes this maximally difficult?

Really, the most interesting questions (from a world-building perspective) are whether it's possible to make this really hard for the inhabitants; and how deep they could be living before pressure from the rock made living impossible.

We could have a rogue planet, cast off from its sun (perhaps by a near-miss with another planet, perhaps by the ancestors of its current inhabitants).

They have dug deeper and deeper as the core cooled. The atmosphere and oceans have been ripped away, If they did dig up to the surface, they'd be dealing with the extreme cold (~3 Kelvin) and hard vacuum of deep space. So they avoid digging upwards. They might have legends of sealing their caves from any fissures that did open, but they come from countless generations back: they are too deep now, 10s of km below the light planet's surface.

The planet is either not spinning significantly, or is tumbling slowly and chaotically enough that pendulums don't help (tangential question: is it possible to make a tumble chaotic enough that you could not factorize it to pitch/roll/yaw with pendulums?)

Losses of knowledge over deep time (plagues, wars, natural disasters) mean that they don't have nuclear weaponry or power, nor can they perform terraforming-scale efforts.

For whatever reason, their records of core temperature measurements go back at most just a few generations, so they can't use the speed of cooling of the core to calculate the insulating thickness of rock above.

(tangential question: How cool would it have to get before there were no longer quakes, and how deep would people have to have dug? Could heat from radioactive material help reduce this? Even the moon has moon-quakes)

Being a rogue planet, asteroid impacts would be too rare to use as stand-ins for seismic effects.

Assuming they don't live in water, their max depth is determined by gravitational strength acting on the rocks above them. On a light planet, like the moon, that could be tens of km.

Could even this deliberately-limited civilization figure it out, then?

Well, divergence of vertical shafts would still give their distance from the center of planetary mass. So they would know their rock universe had a center of mass.

They would be able to measure pressure gradients in the rock, and know the mass of rock, so if they assume similar rock all the way above them, can calculate the depth of rock above them. If they assume a spherical planet, that gives them a planetary radius.

Reflection and absorption of seismic waves would tell them the varying densities of rock on the planet, letting them make the calculations about gravity more precise.

But they can compare that to the gravitational constant to determine that a spherical mass of rock to generate those readings is consistent with the gravitational strength they measure.

And they can use what they know of rock's crush-strength and gravity to calculate the potato radius ( https://www.technologyreview.com/2010/04/12/27697/potato-radius-to-define-dwarf-planets/ ), to prove that the planet is spherical.

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    $\begingroup$ if the planet deviates substantially from spherical symmetry it is possible to tumble chaotically. See unirioja.es/cu/mheras/IJBC132.pdf $\endgroup$
    – lurscher
    Oct 9 '21 at 15:50
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    $\begingroup$ It is reasonable that they could deduce the major inertia moments of the planet with the tumbling patterns, so Focault pendulums wouldn't be useless $\endgroup$
    – lurscher
    Oct 9 '21 at 15:54
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    $\begingroup$ Very interesting :) Just to add to the reasons why they couldn't dig up: perhaps their whole cave system is under an ocean, or some cataclysmic event covered the whole surface in water. Obviously that would require other solutions for oxygen (plants? mushrooms?) but that would allow for a similar reason why they are restricted underground. $\endgroup$
    – Katai
    Oct 10 '21 at 9:11
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    $\begingroup$ AFAIK radioactive heating is a major factor for why the Earth's core is so hot, so presumably even a rogue planet's core would stay warm for a long while (though again AFAIK we have very little certainty about these things even on Earth) $\endgroup$ Oct 10 '21 at 13:39
  • $\begingroup$ Could a planet tumble like this? en.wikipedia.org/wiki/File:Dzhanibekov_effect.ogv $\endgroup$
    – Caesar
    Oct 11 '21 at 9:15
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Others have covered measuring the size and shape of the habitable level. The distance to the surface may be estimated as follows:

  1. Dig a chamber in the rock.

  2. Place a sensitive distance measuring device between the floor and ceiling of the chamber.

  3. Widen the chamber. Since the ceiling now has less support, it will sag, and your measuring device will register this fact.

  4. Seal up the chamber. Pump fluid in until the ceiling height has returned to its original value. Measure the fluid pressure.

In the limit where you start with a narrow chamber and measure a wide one, the pressure of your fluid is simply the weight of the rock above per unit surface area. Since it is assumed you know the gravity and the density of rock, you may thus calculate the distance to the surface.

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    $\begingroup$ This tells you the distance to the surface at the test location. It tells you nothing about whether your planet is a sphere, or has 4000km mountains every two degrees, or anything else about the shape. $\endgroup$ Oct 8 '21 at 20:36
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    $\begingroup$ @RossPresser Others have covered surveying to determine the size and shape, spherical or not, of the habitable level, so I didn't mention it. If the distance to the surface varies, this method, applied at multiple sites, can determine that. $\endgroup$
    – John Doty
    Oct 8 '21 at 20:42
  • $\begingroup$ This assumes that surface rock and deep rock have the same density. Is this true? $\endgroup$ Oct 10 '21 at 13:31
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    $\begingroup$ @RotemShalev In the question as posed, the inhabitants have knowledge of the "Density of surrounding rock and material". Common rocks have densities of 2.5-3, so even without precise knowledge, this method will yield a decent estimate. $\endgroup$
    – John Doty
    Oct 10 '21 at 13:50
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To actually have it occur to these underground people that they live inside a layer of a sphere would take some remarkable insight. Humans began realising this looking at far away things, and got further inspired by looking into the sky (where there are even more distant things!)

IF someone gets the idea though, it's very possible to verify. I can think of two ways to do this on a reasonably small scale.

Find a large underground lake, or construct a long straight tunnel and cover the floor in water. If you lay on the floor at one end, you will find a horizon less than a kilometre away. If you can construct a telescope you could make quite precise measurements, quite possibly more precise than at the surface if you can avoid the wind making waves! Then use the formula in @L.Dutch 's answer.

If you don't have optical instruments or for some other reason find the above approach unsuitable: Find a large cave. The larger the better, but about 250 m high and 1000 might be a minimum requirement. Find some good rope and make a rectangle 250 m by 1000 m, and hang it in the cave such that the sides are vertical. On earth, such an experiment will show that the bottom line will slack more than the top one no matter how you turn your loop, as the difference in length at the top and at the bottom will be a few centimetres. (See the calculation for a bridge here) This would of course be carried out several times, and it requires really precise measurements.

Depending on what materials your people have access to one of the approaches might be more feasible than the other.

Now we know how large the world is at our height! How much world is above us?

I'm afraid this is much trickier.

IF the atmosphere in our caves is connected to the outside, we could measure the air pressure at various altitudes (... negative altitudes?), and then extrapolate to find the height of the atmosphere, but I don't have any good ideas for how to find the amount of rock above you except for the obvious one:

Dig your way upwards!


Bonus answer, if you're on a (very) small planet: One experiment which is marginally easier than to simply go around the world is to make a large triangle. An equilateral triangle in a plane will have all angles equal 60°, but an equilateral triangle drawn on a sphere will have larger angles. Using the spherical law of cosines (https://en.wikipedia.org/wiki/Spherical_trigonometry) one finds the angle $A$ of a spherical equilateral triangle with side $s$ equals $$ A = \arccos \left( \frac{\cos x - \cos^2 x}{\sin^2 x} \right) $$ where $x = \frac{\text{Triangle side length}}{\text{Planet diameter}}$. This might actually work on a small planet or moon. On earth, an equilateral triangle with side length 2 000 km has angles of about 61°, so don't try this at home.

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Once you know that your planet is spherical and have calculated it's size using observational trigonometry as outlined in the other answers you can calculate the approximate size of the world and thus your depth with a little physics.

Gravity at a depth below the surface varies due to less mass below you and the pull of the mass above you. A related Physics SE question relates https://physics.stackexchange.com/questions/18446/how-does-gravity-work-underground

If they have the ability to measure the local gravity at different depths to great enough precision they could calculate the mass of the planet and it's radius at the surface. Density variations would give some variability, but it would give a first approximation and could be refined based on increased geological knowledge of their planets interior.

EDIT:

To clarify how this works, a rough equation for gravity at depth d is:

g(d)= G x M x (R−d)/R^3

Where G is the gravitational contant, M is the mass of the planet, and R is the radius of the planet.

If G and M are known and gravity can be measured directly at 2 depths a known distance apart you could solve the two equations for both the radius of the planet and the corresponding depth of one of the gravity measurements.

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  • $\begingroup$ It's not clear to me how this method lets you know if you're near or far from the surface. Suppose you find some relationship of gravity vs depth - I don't see how you can extrapolate that to find the location of the surface unless you know either the strength of gravity on the surface (which you wouldn't), or the thickness of the different density layers above you (in which case you already know the size of the planet). $\endgroup$ Oct 8 '21 at 18:58
  • $\begingroup$ Edited the answer to explain more in depth how it would work. $\endgroup$
    – Josh King
    Oct 8 '21 at 19:09
  • $\begingroup$ But how do you find the mass of the planet (M) without knowing how big it is? $\endgroup$ Oct 8 '21 at 19:16
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    $\begingroup$ It's all systems of equations algebra, take a third measurement of gravity at another known relative elevation and you would get a third equation to solve for the third unknown (planetary mass) It all depends on accurately measuring gravity $\endgroup$
    – Josh King
    Oct 8 '21 at 19:50
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    $\begingroup$ By the Shell Theorem (en.wikipedia.org/wiki/Shell_theorem), the gravity of a uniform spherical shell is zero in its interior. You thus cannot measure the thickness of a spherical shell by measuring its gravity from the inside. $\endgroup$
    – John Doty
    Oct 8 '21 at 19:57
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Foucault pendulum was already mentioned. 3 more ways to detect rotation of the planet with simple devices, taken from MOTION MOUNTAIN by Schiller.

In 1913, Arthur Compton showed that a closed tube filled with water and some small floating particles (or bubbles) can be used to show the rotation of the Earth. The device is called a Compton tube or Compton wheel. Compton showed that when a horizontal tube filled with water is rotated by 180°, something happens that allows one to prove that the Earth rotates. The experiment, shown in Figure 104, even allows measuring the latitude of the point where the experiment is made.

enter image description here

In 1910, John Hagen published the results of another experiment. Two masses are put on a horizontal bar that can turn around a vertical axis, a so-called isotomeograph. Its total mass was 260 kg. If the two masses are slowly moved towards the support, as shown in Figure 103, and if the friction is kept low enough, the bar rotates. Obviously, this would not happen if the Earth were not rotating.

Hans Bucka developed the simplest experiment so far to show the rotation of the Earth. A metal rod allows anybody to detect the rotation of the Earth after only a few seconds of observation, using the set-up of Figure 106. The experiment can be easily be performed in class. Can you guess how it works?

enter image description here

A usable gyrocompass was invented in 1906 in Germany by Hermann Anschütz-Kaempfe, and after successful tests in 1908 became widely used in the German Imperial Navy.

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It Would Be Difficult to Prove

One situation where I don’t see how is if they start in the center of the planet. The gravity on all sides cancels out, so they're weightless and a plumb line won’t lead anywhere. Therefore, none of the answers that say they’ll follow a plumb line will work.

Digging up in any direction will reveal that gravity is a force that pulls them back to the center of the universe, and gets stronger the further away they dig. If the planet is rotating, they can measure that too, and see it gets stronger the further away they get.

But the obvious, intuitive explanation for this is that they live in a cosmos of solid rock around their home in the center of the universe.

But Maybe They Could

They could in theory dig all the way up to the surface, or the impenetrable crust, but you say these people do not have that ability.

They can figure out that gravity and the rotation of the universe can’t get infinitely strong, so someone probably does theorize that there’s some distance where the forces on rock must pulverize it, and, in theory, the rock within that radius would form a sphere of solid rock covered in dust. They know the rate at which gravity increases and the amount of force it takes to crush the kind of rock they've been digging through.

You say they can know the gravitational constant. They can precisely measure the gravitational pull of things like dense mineral deposits versus pockets of gas. It's at least possible someone might come up with the hypothesis that gravity is caused by there being more mass below them than above them, and realize that this matches the gravitational pull (and rotation, if they are within a rotating planet) of digging up through a sphere of uniform density. This would let them estimate the radius of a sphere of uniform density required to explain the change in gravity, although they have no way to know if the rock gets denser or less dense higher up without digging through it. It would be difficult to rule out other hypotheses, but @10ebbor10 suggested they could detect the reflections of seismic waves from the surface.

Modeling their world as a sphere of uniform density, and knowing the density of rock and the gravitational constant, they can calculate the gravitational pull of the mass beneath a point a certain distance from the center and subtract the gravitational pull of the mass above with a bit of integration in cylindrical coordinates. Because of symmetry, all lateral forces would cancel out. Measurements of the force of gravity at different heights would then let them solve for the radius of the sphere.

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