1) We can make very precise very frequent observations of the motions of the planets (which move differently from stars) and the Sun over a long period of time, which allows us to trace their exact paths to a very high degree of accuracy, and everything in the solar system matches perfectly a heliocentric model where planets orbit in slight ellipses around the Sun, and when we take into account the Sun's apparent movement on the sky, we can see that it matches perfectly what we would see if the Earth was also orbiting around the Sun, just like the other planets.
This is not just mere claims, this is not speculative: This is actual real-life empirical observation and measurement to high degrees of accuracy, at frequent intervals over very long periods of time, and it matches perfectly. And not only that, but it matches perfectly what the orbits would be if we assume that the Sun exerts the same gravitational force on the planets as we can observe right here on Earth, with practical experiments and measurements.
(Granted, this all in itself does not yet tell us for certain that the Earth is a globe, but it gives us an extremely strong hint of it because we can observe all the other planets being globes, and the Earth appears to be yet another planet among them, so there's no reason to not believe it's also a globe like the other planets.)
2) Science can predict solar and lunar eclipses with extreme accuracy, both in terms of when they will happen, and the exact path that the shadow will make, decades and even centuries in advance, using the exact same solar system motion mechanics as described above. This is not some kind of magic or satanic power, it's pure observation and math, and is 100% consistent with the heliocentric model.
Flat-earthers have literally no explanation to why science is able to predict eclipses with such enormous accuracy so long in advance.
3) The fact that the Earth is just another planet, like the others, means that it's rotating (rather than the Sun orbiting around the Earth), and we can actually measure this rotation. Modern instruments can measure this rotation rate with extreme accuracy and precision, without leaving any doubt that we are, indeed rotating. This rotation can be measure everywhere on Earth, and its rate and direction is always consistent with us being on the surface of a sphere rather than a flat disc.
4) Likewise, the same precision instruments can be used to measure the change in orientation as we traverse across the surface of the Earth. Indeed, the famous "8 inches per mile squared" can be literally measured by how much your orientation changes as you traverse several miles. This rate is the same everywhere and consistent with the change in orientation on the surface of a slightly oblate spheroid.
5) We can measure distances between different parts of the world with high accuracy. We have been able to do this with decent accuracy for literally centuries, but obviously with modern technology the accuracy has only increased. And every distance we measure between two points on Earth (for example two cities), no matter where they are, very much including those points on the southern hemisphere, correspond exactly to all those measured points being on the surface of a sphere (or, more precisely, a slightly oblate spheroid).
Travel times by car, sea ships and airplanes are completely consistent with those distances, regardless of which two points on Earth we are talking about.
6) We can measure the apparent altitude of the Sun on the sky at the same time (eg. midday), and this apparent altitude of the Sun is always consistent with the latitude where the observation is made from if we were on the surface of a sphere, and inconsistent with the notion that we are on the surface of a flat disc. Same thing when measuring the altitude of stars in the sky. (In fact, measuring the altitude of the Sun and stars has been used for navigation for centuries, and has always been consistent with a globe Earth.)
7) The apparent movement of the Sun in the sky is consistent with us being on the surface of a rotating sphere, not with us being on the surface of a stationary flat disc with the Sun moving in circles above it.
(One example of this is that the Sun always raises almost directly from the East, not the North-East, which should be the case if the flat Earth model were true. Likewise it sets on the West, not the North-West. This is something that can be very easily measured with common household objects. You don't have to just rely on someone's claims.)
The same goes for the stars, at every point on Earth.
8) Unlike flat-earthers claim, you can directly witness eg. sea ships disappearing beyond the horizon from the bottom up, and this will be so regardless of what kind of photographic equipment or telescopes you use. You will not be able to "zoom it back into view" when the image is sharp and the ship has traveled far enough. There are also other structures where this curvature away from the viewer can be clearly seen, such as certain very long straight bridges.
9) Circumnavigation of Antarctica is perfectly possible, and takes the amount of time predicted by the globe Earth model and ship speed, and would absolutely not work on the flat Earth (where it would take enormously longer).
And while there, the Sun and the stars behave exactly as you would expect if the Earth was a rotating globe.
10) GPS would categorically not work as it does on the flat Earth model. This is something you can check and research yourself, first-hand, rather than rely on someone's word on some measurement unattainable by regular people.
And those are just a tiny fraction of all the ways in which we know, with absolute certainty, that the Earth is a globe and the solar system is heliocentric, and it's not even going into the fact that pretty much none of the science that we have enormous amounts of practical uses for would work in the flat-Earth universe.
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