How to prove the earth and moon isn't flat?

Except that is begging the question. Eratosthenes assumed the Earth was round and didn’t prove it. The same results (7 degrees 5000 stadia away) can be achieved with the Sun as a point-source approximately 40720 stadia from the flat Earth.

While we currently think of the Equator as the place that’s equidistant from the poles, it’s original meaning was the place where day and night are always equal.

Separately: I have a house on the ocean that sits on a bluff. On a clear day, we can see the horizon to the east. I have one child standing at the shore and the other standing on the bluff about 50 feet away and 20 feet higher. I tell the kids to yell when they first see the sun. The one 20 feet higher can see the sun a few seconds before the one standing at the water’s edge. I tell the kids that this proves the earth is not flat.

I was thinking about this question and I think the OP has a few tacit assumptions that he may not have intended but certainly would make many of these solution impractical. Remember we are talking thousand of years ago (maybe a different scenario would be modern world finding a primitive tribe).

Equipment: What is the error/impreciseness of the equipment? Remember - no modern clinometers or theodolites. At best you can make a rudimentary quadrant or sextant (so no time-lapse photography). Does interpretation of the results involve knowing/learning trigonometry or at the very least geometry?

Travel: Unless you took a car back too (Quercus I’m looking at you) then you have to walk 6.67 miles to achieve 1 degree of elevation difference. Is that within equipment tolerances or would they understand the implication (see Equipment). Even so, I can write a trig equation for the results that assume a flat Earth. You would have to take at least three measurements so the distance between 1 & 2 and 2 & 3 are spaced equally apart and know (or prove) that tangent is not a linear function.

Location: Are you close to a coastline? If so, does the culture have long-range boating. If not, is there a variation that would work? Desert maybe but what about mountains or forest?

Modern conceptions: Mangetout assumes timezones are obvious. Then how come it took until 1883 and the advent of transcontinental railroads before time zones came into existence? Time zones simply don’t make sense unless you can travel east/west fairly quickly.
So let’s throw this back on the OP. How far are we allowed to travel? Assuming we can make equidistant marks on an arc, do you allow us a sextant out of sticks and shiny rocks? Or a clinometer out of a sight and string and a weight? If so, what is the error? 1 degree? 2 degrees? Can we throw out any answer that assumes travel if the two answers contradict each other?

Does it? If I assume the Earth is flat and the horizon is the edge then someone at elevation will see the Sun earlier. Thought exercise. Take a flat plate 1" above an infinite table and put a camera on the plate and the other some height above the plate (since the table is infinite it does not matter how high) both pointed to the edge of the plate. The camera on the plate cannot see the table since it lies parallel to the table but the camera at elevation will. Does this prove the plate is curved?

No it does not. Same answer as that to bizerta. In fact I’ll elaborate on that answer. Remember the plate? It has a radius of 6" and even closer to the table - 0.01" to simulate that the Sun has just set. A lightbulb is set in the table right at the surface. Assuming you’re anywhere on the plate (except the edge) it is impossible to see the lightbulb no matter how far away it is. Assuming you’re at the middle of the plate and slightly above it, it is easy to show that if your distance above the plate times the distance of the light from the edge of the plate (both in inches) is larger than 0.06 then you can see the light.

Hmm, alright. So, what I have been personally trying to accomplish with this thread is to be able to have a practical solution to proving the earth and moon are spherical. As of now the time traveling idea seems to be a stumbling block to that.

I apologize to those who invested a lot in the time traveling idea. I didn’t want to say the above at first because it seemed to be the primary theme of this thread, and I didn’t want people to think I was wasting their time. I thought I would possibly still be able to get my answer, but at this point the time-traveling scenario is running counter to my personal goal of this thread.

Let’s say at this point, anything goes (in terms of coming up proof for a spherical earth and moon), but the more practical the better (e.g. making calculations based off of the stars before going up in a rocketship to space to look at the earth).

Sorry for that. Feel free to go on with the time traveling concept if you wish, but unfortunately that is not what I’m looking for.

Build a Foucalt pendulum.

Beyond that, we know that the gravitational attraction of the sun and moon makes water “pile up” to various degrees, even when the sun and moon are on the opposite side of the earth, so there’s no reason to think it wouldn’t do the same if the sun and moon were revolving around a flat earth. Maybe the people seeing the sun first from the cliff are just looking over the hump in the water.

Eratosthenes’s method for doing this around 200 BC was to erect two sticks of identical length, and plumb to the earth, hundreds of miles apart (the aforementioned “gnomon”) and to measure each of their shadows cast by the sun at a specific time on the same day, then comparing notes.

If the earth were flat, the length of the cast shadow would be the same for both sticks. But if it’s spherical, the two shadows will differ, and would offer proof: I would demonstrate this by building scale models, for the skeptics, of a flat and spherical earth with two sticks in each, having their shadows being cast by the sun itself. The correlation to a spherical earth should be convincing.

You could use a diagram similar to that here from xkcd’s blag, to illustrate the implications of this experiment:

Go to Egypt, get two other people, keeping one on the desert floor of the Great Pyramid of Giza, and the other at the pinnacle of the pyramid.

Each of them have a bright red flag. They are instructed to wave it as soon as they see the upper limb of the sun disappear beneath the horizon.

If the earth were flat, both would wave their flags simultaneously. Of course, since it’s not, the man on the ground would wave his flag before the woman at the top waved hers a bit later.

Only if the sun were infinitely far away.

Hold a lamp a couple feet above a flat table. Put one vertical stick right under the bulb; it casts a short shadow. Put the same stick, also vertical, a few feet away from there, and it casts a longer shadow.

If you know that the sun is 93 million miles away, then you can prove that what I just described can’t account for the difference in shadow lengths measured by Eratosthenes. But how do you prove to Og and Bog that the sun is 93 million miles away?

93 million miles is so far away, that the parallax this causes is so tiny, it is well within the margin of error for this purpose, and practically speaking, may as well be an infinitely far light source.

You do realize you completely missed the point of what TonySinclair was saying. His point was that the different length shadows can be because of a very far away sun and a round earth or a much closer sun and a flat earth. How do you tell the difference?

People did not decide the earth was round because of one type of observation. They decided it was round based on many different sorts of observations. Each of which had many possible causes but when taken in aggregate make a spherical earth a plausible explanation.

I did miss that point. A good one at that.

I suppose at some point you’d have to give the ignorant a crash-course in Trig, and also show the sun to be 93 million miles away. Aristarchus did this just a handful of years before Eratosthenes used the gnomon method.

Also, here’s Huygens’s method.

You’ll note in my post about it, I went further and calculated the distance to the Sun assuming a flat Earth.

Absolutely wrong. A person at elevation can look over the lip of the flat Earth and see the Sun earlier. Do this. Draw a segment (representing the flat Earth and label the endpoints A and B. Draw the perpendicular bisector and label the intersection M. Now draw a point on the bisector above M and call it N.
Draw ray NB and label any point past B as O.
Extend the segment AB past B to make it ray AC.

If the sun is anywhere in the angle CBO then person at point N sees it and a person at point M does not.

This can be generalized to any perpendicular to segment AB except at the endpoints.
Therefore the Earth is flat.
Quod Erat Demonstrandum
Oh and by the way on the real round Earth - the person on top of the pyramid sees sunlight before the person on the ground - not the way you describe it.

It might be worth considering some actual historical evidence here.

We can date when expert opinion went from believing that the world was flat to believing it to be a sphere to within about a century and a half. The Greek philosopher Anaximenes, who lived c. 585 – c. 528 BC, clearly believed it to be flat, a did his probable mentor Anaximander (c. 610 – c. 546 BC). (I mention Anaximander because he is credited with having created the first world map - no doubt very inaccurate - and incidentally, with introducing the gnomon to the Greek world.) On the other hand, Aristotle (384 – 322BC), born about a century and a half after Anaximenes died, not only holds the Earth to be a sphere, but clearly thinks that opinion to be quite commonplace and uncontroversial. So the discovery came somewhere between about 528 and 384 BC, most likely toward the earlier end of that period.

Unfortunately, we do not have direct evidence of who discovered it or how. (Or, more to the point, who it was who persuaded the majority of his peers that it was so.) The best evidence we do have comes from surviving fragments of the writings of Theophrastus, Aristotle’s star pupil, who wrote the first systematic history of philosophy. Theophrastus attributes the discovery to Parmenides (of rather uncertain dates, but probably born somewhere between 515 and 540BC). I believe his method of proving it was thought to have been that mentioned by AncientHumanoid: he observed the shadow cast on the Moon by the Earth during a lunar eclipse, and compared the shape of the terminator with the shapes of shadows cast by variously shaped objects on one another. He found that it best matched teh shape of a shadow cast by a sphere upon another sphere.

It is worth mentioning the many ancient authors actually attributed the discovery that the Earth is a sphere to Pythagoras (and I think there may be a few attributions of the discovery to other figures of that , such as Anaxagoras - although he almost certainly lived much too late). However, all these attributions to Pythagoras come from long after Theophrastus’ time, and modern expert opinion is that they are almost certainly spurious. Pythagoras did not do it. In fact, Pythagoras, about whom very little is known for certain, had become a semi-legendary figure by late antiquity, and all sorts of opinions and discoveries (including “Pythagoras’ theorem”) were falsely attributed to him. In fact he was probably more like a religious prophet of some sort rather than a philosopher, scientist, or mathematician, although it does seem to be the case that some of his followers, a generation or two after his death, did become very interested in mathematics, and did discover the theorem, amongst other things.

Unfortunately this scholarly consensus that most of the discoveries (and doctrines) later attributed to Pythagoras were not really his has only emerged quite recently (see here), and much older scholarly material, and even quite recent secondary sources, (let alone much stuff on the internet) continue to repeat the old, exploded stories about him.

Aristarchus’ estimate of the distance of the Sun (or, rather of the relative distances of Sun and Moon) was however, hugely inaccurate, whereas Eratosthenes estimate of the size of the Earth was amazingly good. Aristarchus estimated the Sun to be only about six times as far away as the Moon is. His mathematical analysis of the problem (which does not require trigonometry of the modern type - which was not developed until many centuries later) was sound. However, his measurements were not so good, and this is one of those situations where a tiny error in measuring can lead to a huge error in the final result.

In any case, Aristarchus, like Eratosthenes, was living at a time when the sphericity of the Earth had long since been taken for granted.

The way an ancient Egyptian figured out that the earth was round (and calculated the diameter as well as anyone for a thousand years or so) was to read the reports from soldiers stationed at various obelisks.
Among the reports were date and time (a neat trick for the age) and the distance of the shadow cast at noon.
One reported a shadow, another, some non-trivial distant, reported there was none.

The sun was striking the towers at different angles at the same time.

There were few other explanations.

To boot, he actually hired a man to pace off the distance between the obelisks - that distance + the angular deflection indicated by the different shadows allowed him to calculate the diameter.

Compared that trick, the bit of telling the boss when the Nile would flood each spring (so he could “order” it to flood) was easy.

Do you have a cite for any of that?

I am pretty sure that the “ancient Egyptians”, the civilization that built the pyramids and was ruled by the Pharaohs, believed that the Earth was flat.

On the other hand, Eratosthenes, who has already been discussed quite a lot in this thread, and who did measure the size of the Earth quite accurately, did live and work in Egypt in a later era, although I believe he was ethnically Greek and was born is Syria, and belonged to the Hellenistic culture which owed more to Greece than to Egypt. Probably you are thinking of him.

Eratosthenes did measure the Earth by measuring shadows at noon at different places, but it is news to me that this involved reports from soldiers about obelisks. In any case, as has already been mentioned several times, he did not discover that it was a sphere. That had already been known for about two and half centuries by his time.

I think that would demonstrate that it’s in motion, but not its shape.