Now, when i say proof, im being careful not to conflate it with evidence or demonstration; proof as in 100%, non subjective, not up for interpretation,
mathematical, syllogistic, proof.
This relies on two very simple, I'd imagine indisputable, premises, and one trivially easy to confirm premise:
1) The path the sun takes in the sky, when viewed from earth, is a
relatively regular circle. In about 24 hours the sun will be in the same spot it was yesterday in the sky, with no teleporting, drastic speeding up, slowing down, or overt change in direction.
2) The equator is also a mostly circle shape (considering the equator could be defined as the path which the sun travels directly overhead on both equinoxes), with no particular beginning or end.
Those two I don't expect anyone to disagree with. Neither need be a perfect circle, just reasonably describable as a circle.
3) Twice a year, from every point on the equator, during the spring and fall equinoxes, the sun travels in a path directly overhead in a perfectly straight line, east to west.
The FAQ says you use an empirical approach and rely on your own senses, so I would be surprised if any of you disagreed with this premise.
For clarity, my premises restated and truncated:
P1) The sun travels in a circular path
P2) The equator is a circle.
P3) During the equinox, from any and every point on the equator, the sun appears to travel in a straight line.
C) The path of the sun and the equator are on the same plane during the equinox.
NOTE: this proof pre supposes no world model shape and needs none to reach it's conclusions.Now, mathematically, there are exactly three ways to arrange these two circles so that one appears as a straight line from every point on the other:
Both on the same plane, laying next to each other as so:
OOI think we can agree that that can in no way describe the path of the sun.
The only other two possible configurations are for both circles to be on the same plane, either the path of the sun iside the equator, or the equator inside the path of the sun. They would also be concentric circles (if they aren't concentric, then the sun would be closer to the equator at some point during the equinox, a phenomenon I think we can agree doesn't happen). I think we can agree that for the sun's path to be the inner circle (a concave earth with the sun on the inside, viewable from every point on the equator at all times) is ridiculous, so that leaves us with exactly one possible configuration:
The path of the sun is the larger of two concentric circles on the same plane, with the equator being the inner circle.A viewer from any point would be standing on the equator, looking up, their head pointed directly toward the path of the sun, the outer circle, directly away from the center of the circles. His head would be pointed away from the center of the circles, his feet pointed toward the center of the circles. He would also be oriented on the plane of the equator-sun system as if he were lying on it, though he is in fact standing. The sun would start in the east, travel in a straight line until it is overhead, continue in a straight path down toward the horizon, continuing downward. Being on a single plane, it can not turn, so would continue in a straight path down below the horizon. If he stood still, 12 hours later, the sun would be directly under his feet, an a viewer at the opposite end of the equator would be looking up directly at the sun. Their feet would be pointed toward each other, they would each think the other is upside down. Down for each of them would be toward each other, toward the center of the circles. in fact, every person on the inner circle would see 'up', directly toward the apex sun, as away from the center of the circles, and down as toward the center. Because this is describing people encircling a three dimentional Earth with a perceived downward force toward the Earth's center, the earth can therefore not be flat.
For a more hands-on model of the same proof, one that you need not be on the equator or during an equinox for, take a hula hoop and mark two opposite ends with tape (a trick is to mark one end with tape, hold it up by that end with one finger, and hang a coat hanger on the other end, the hanger will slide to the exact opposite end). Denote it as you so please, but one half will represent the day, the 12 hours we can see the sun during the equinox, and one half will represent the night, the 12 hours we can not see the sun during equinox. The tape will represent sunrise and sunset, 12 hours apart from each other on the sun's equinox path. Now hold the hula hoop in such a way that both pieces of tape lie on the horizon from your point of view(as the sunrise and sunset would), and the day part of the hula hoop is directly overhead from your point of view (as the midday sun would be(depending on the size, you may have to bend your torso in silly ways)). You need not orient east / west for this experiment. The hula hoop is relatively a circle, just like the path of the sun. The important part to notice is what happens to the 'NIGHT' side of the hula hoop if you align the sunrise and sunset at the horizon and the midday sun directly overhead. The path of the sun can only pass below you, your local midnight it would be directly below you, directly above someone else on the opposite end of the equator, who would be relatively upside-down to you.
I curious what you think. I'd love feedback on how to clarify what I'm saying or someone pointing out where some of my logic is flawed, though, it's syllogistic logic, so I'm not sure where I could have possibly erred.
Thanks for your time.
Edits: minor grammar, spelling, and phrasing for clarification.
Edit 2: Image added and some formatting to address Mysfit's confusion.
Edit 3: more formatting and rephrasing, and a correction to address Mysfit's valid criticisms.