In the video below a man measures the apparent height of a lighthouse from near to sea level and then from the top of a cliff somewhere in Australia. The view from the top of the cliff allows the base of the lighthouse to be seen whereas the sea level view does not.
The two viewpoints are then checked against Google Earth which uses a round Earth model of radius 6400 km. The views are a good visual match and so we can say that the observations are consistent with the Earth being curved at that location and has a curvature close to the standard model as explained in textbooks.
There was, once upon a time a similar video made by someone in the UK but it appears to have vanished from YouTube. So if the Earth is curved in both Australia and the UK then it is a fair bet that it meets up somewhere in between and is in fact a globe.
A hundred or so of such experiments with multiple auditors would surely settle the matter once and for all.
The Bedford Level Experiment is often cited by Flat Earthers as proof that the Earth is flat but the account on Wikipedia does not support this. A telescope was sighted along a ten mile canal and a boat was viewed sailing into the distance. When the eye-line was close to the water the boat stayed visible and this was claimed to be proof of a flat earth.
Surveyors, however, claimed that this was because of the refractive properties of the air close to the water (the light is bending around the water a little) and that this was something that they regularly took account of in their daily jobs.
Further experiments with a higher eye-line and with a pole placed halfway along the canal gave results consistent with a curved Earth of the correct radius. The midway pole appeared to be higher than the line joining the poles at each end of the canal – see diagram.
These two experiments and others of this type are really the most rigorous way of showing a curved Earth as they make no extra assumptions concerning other factors such as the movements of stars and sun or the position of the Earth in space.
A statement such as “The Earth is a globe” or “The Earth is flat” are statements purely concerning geometry and hence demand purely geometric explanations.
Eratosthenes of Cyrene (276 BC – 195 BC) worked out the different angles of the midday sun at Syene and Alexandria by measuring the length of the shadows of two vertical poles placed at those locations.
From this he calculated the circumference of the Earth as 25000 stadia which depending upon how large you think a stadion is, gives an accuracy of 15%, 10% or 0,16% as compared to modern estimates.
Eratosthenes’ calculation depended upon the assumption that the sun was a so far away that the rays coming from it were effectively parallel. Flat earthers have seized upon this by pointing out that a sun which was much nearer in combination with a flat Earth could give the same result.
So far they are correct but the argument breaks down if three or more measurements are made. In this case we would always get different results for a flat and round Earth models. This is therefore one way to distinguish easily between the two ideas; just repeat the experiment with more poles.
There are plenty of other arguments that suggest a round Earth but all involve the introduction of extra variables such as spinning Earth, rotating stars, sunsets, planetary orbits, gravity, eclipses and seasons etc. The arguments are correct but have little to do with the shape of the Earth and each extra element gives the flat Earthers something else to argue about.
Having said this though, the real reason that most scientists think the Earth is round is precisely because of these extra variables, they all fit together into a coherent mathematical model of the solar system that can be used to make very accurate predictions of future cosmic events.
The complexity of the model is such that each part of it supports all the other parts, meaning that if any one part is wrong then probably all parts are wrong. Furthermore, the complexity is such that few laymen can understand it and there appear to be no easy tutorials available, in contrast to the flat Earth model where there are many well produced video tutorials that lead the viewer expertly down a series of enticing but misleading rabbit holes.
Modern scientists have been very lazy with respect to providing proof of round Earth that is understandable to a layman (we are reduced to citing an ancient Greek!) but there is enough evidence here to favour a round Earth interpretation over a flat Earth model.
We have several experiments that suggest the Earth is round and all give very similar estimates for the radius, which seems very unlikely to be a coincidence.
Bedford Level Experiment – Wikipedia
The History of Geodesy (NOAA) – Global positioning tutorial