I'm sorry if this isn't what you want to hear, but there isn't a single piece of argument for a flat Earth that can't be easily debunked with a modicum of sound scientific knowledge.
When the I first heard about the flat Earth conspiracy theory, I decided to try to take the time to objectively analyze every alleged piece of argument for flat Earth I stumbled upon, partly for intellectual honesty but mostly just for fun, I have to admit.
We can't convincingly refute unsound arguments if we don't understand them.
Contrary to what most conspiracy theorists believe, the scientific community at large would be actually delighted if there was any tangible proof that Earth is flat. So much new science to work on! But to this day I haven't come across anything that ever made me doubt that Earth is indeed a very, very slightly oblate spheroid. It's so close to a perfect sphere that if it was shrunken down to the size of a billiard ball it would be within regulations.
I find it quite fascinating and slightly depressing that so many seemingly intelligent people are attracted to such bizarre and erroneous lines of thinking, instead of sticking to Occam's razor. If you think about it, the elucubrations of flat Earth hypothesis proponents are actually a regression to pre-Ptolemaic astronomy, deferents and epicycles included.
That said, there's one thing I can affirm with absolute certainty.
There isn't a single flat Earth model that explains at the same time two simple phenomena we can all directly and consistently observe ourselves :
- The fact that at any moment in time exactly half of Earth is illuminated by the Sun and the other half is dark. A few well timed phone calls to a group of similarly inquisitively-minded friends living in different time zones can quite easily ascertain that.
- The fact that we experience the passing of seasons through the year and that they're the opposite in the Northern and Southern hemispheres.
If
someone tells you they have proof that Earth is flat, insist that they
show you a scientific model accounting for both of these phenomena at the same time. They won't be able to do it. No
flat Earth proponent ever could, because it simply doesn't work without
the two separate tentative models contradicting each other.
That's a definitive nail in the coffin of the flat Earth hypothesis, while a heliocentric globe Earth model accounts for both of these basic observations perfectly well, at the same time.
On the other hand, it's very difficult to explain something as simple as this, for example:
or this:
People in general, not just flat-earthers, often seem to fail to grasp how big and massive Earth actually is, compared to the scales our brains are wired to deal with, and that's perfectly normal.
When we think about it, most of us picture in our heads something about the size of a basketball.
That's a scale our brains can comfortably manage.
But try instead to ponder about the fact that Earth's circumference at the equator is about 24,901 miles (40,075 km) and that its radius is about 3,963 miles (6,378 kilometers).
Two friends living 620 miles (1000 Km) apart on a North-South axis, each equipped with a two feet long straight piece of broom handle, a carpenter’s plumb, a watch, an optional mobile phone and some basic geometry knowledge can ascertain that too.
School children all over the world conducted this 2205 year old experiment to calculate Earth’s circumference
Then consider the fact that for an average 5'7″ (1.70 m) tall human observer standing with her/his feet in the water at any beach on Earth, the distance to where objects start to disappear behind the horizon between sky and water is only about 3.1 miles (5 Km).
When flat earthers insist that “we can easily observe that the surface of water is level, hence Earth must be flat”, it's actually just a problem of scale.
The surface of say a bucket of water, or a swimming pool, is indeed for all intents and purposes flat on a small scale, locally.
The Earth's curvature “bulge” over the size of an Olympic swimming pool is about 1 micron (0.001 millimeters, or 0.04 thousandths of an inch) in the middle.
That's indeed very flat, even for a mechanical engineer.
But on larger bodies of water the curvature becomes incontrovertibly apparent and measurable.
Over a distance of 5 kilometers (3.1 miles), Earth's curvature “bulge” on a body of water is about 2 meters (6 feet and 7 inches), and it's possible to observe and measure it empirically, if the proper precautions to avoid the effects of atmospheric refraction and temperature inversion are taken.
Fun fact, the first well-documented experiment to visually measure Earth's curvature, or actually in this case supposed flatness, was conducted by the British flat Earth proponent and founder of the original Flat Earth Society Samuel Rowbotham
He claimed to have proven that Earth is flat in 1838, but the experiment was repeated several times by different people, with contradicting results, because refraction wasn't always properly taken into account.
The full story is actually quite hilarious: non-honored bets, insults, slander, lawsuits and public death threats