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Anybody who gives the flat earth idea the slightest amount of critical examination will realize that it's complete balderdash.  
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Saying that 6 miles at sea should account for 24 feet of drop is nonsense: a sight line 6 feet above the water would meet the top of another 6-foot pole at 6 miles' distance if the earth were "flat."
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Here is a video of a large ship quite clearly going over and behind the curvature of the sea:
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https://www.youtube.com/watch?v=mUgKxxR9XkU
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The container ship starts quite visible with most of its hull in view, but sailing away for an hour appears to sink gradually into the sea.
Eventually the containers on deck and the ship's superstructure are all hidden behind the apparently rising water level of the sea.
The smaller boats in between the container ship and camera are not rising and falling over huge waves, so the swells are quite small, about 2 feet.
Can you refute this simple evidence?
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The website page below compares views of Polaris from various latitudes on a "flat" earth vs. a spherodial earth.
The angle of sight from earth to Polaris is identical to the viewer's latitude on earth!
All viewers are seeing the sun at the same time, so the sun must be located in the same place, consequently all viewers must be looking in the same direction to see the sun there.
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Does this support flat-earthism or is it rather supportive of a spheroidial earth?
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http://i.imgur.com/CPU63Tm.jpg

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Can you refute this evidence that the sun is much further away from earth than the moon is?  
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https://www.youtube.com/watch?v=lVgx0Eio2Mg
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An excellent response to flat earth nonsense. 
Numbers and simple math disproves all their childish yammering.
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It's painfully obvious that the sun doesn't twirl around above the earth the way flat-earthers claim.
How would Antarctica get 24 hour sun if it did?
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https://vimeo.com/136977957

And no, Polaris has not always been the Pole star in the past, nor will it be in the future.
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Essentially, belief in the silly idea the earth is "flat" relies solely on ignorance.
For at least some, their ignorance is willful and impudent, therefore quite repulsive.
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Anybody who gives the flat earth idea the slightest amount of critical examination will realize that it's complete balderdash.  
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Saying that 6 miles at sea should account for 24 feet of drop is nonsense: a sight line 6 feet above the water would meet the top of another 6-foot pole at 6 miles' distance if the earth were "flat."

There is no curve to the sea because water surface doesn't magically hump-up higher in the center between two shores, nor does the surface of water ever curve.  Water can slosh, but it always settles flat by nature.  Everyone knows that 'sea level' is the horizontal foundation of measurement used in determining elevation of land height.  It's how we know Mt McKinley is 20,310 feet tall. It certainly isn't a 'sea curve' based measurement, or there could be no certain measurement regarding the height of land.  Level is level on a flat earth.  Curve is curve.  But on a globe, you relativists insist the terms are interchangeable, a proposition without foundation.  Earth is not a globe.    

I hope your tendencies remain consistent with the Dogmatic Flatearthists on this site.

For both sides, I am trying to take the evidence and examine it on its own.  Just because I dismiss Neil as a biased source doesn't mean I don't look at the things he posts.  It's just very difficult to filter out the core factual content when it's surrounded by 3 or 4 paragraphs of emotional ranting and insults.

I'm considering the picture he posted in the OP to understand what its implications are.

What I don't like is when people ignore something and then post OTHER evidence either unrelated or only tangentially related to it ... without actually addressing the main point.

So, if someone posts a picture which appears to show curvature, then someone else posts a different picture that shows no curvature.  If both pictures are genuine, then there must be an explanation for why one of them isn't proving what the poster claims that it does.

So, for instance, a globe earther will show video of a boat getting lower and then disappearing as it moves away from the observer.  But then flat earthers have shown convincing evidence that this can APPEAR to be the case even when it really isn't ... if you zoom in on the boat with some magnification.  So they have demonstrated that the visual appearance of something sinking beyond the horizon doesn't prove globe earth ... since it can just be an optical phenomenon.  So this is the kind of analysis I'm trying to do ... amid all the emotional noise on either side.

Neil at one point posted video of SpaceX as if it proved globe earth, but then flat earthers provided convincing evidence that the pictures were fake/CGI/something and could not have been genuine.  Does that prove flat earth?  No.  But it showed that that particular piece of evidence from globe earthers was invalid.

Now, on the other side, I've seen pictures taken by objective third parties of mountains that could be seen from WAY TOO FAR AWAY given curvature math.  These were from people who had no interest in disputing globe earth.  I've seen pictures of other objects that would not be visible at the stated distances.  And there's only one explanation from globe earthers ... refraction.  If you're a globe earther, you assume that there MUST have been refraction (because of your premise that the earth is a globe).  If you're a flat earther, you assume that there was NOT refraction involved.  If you're trying to find the truth, then you sit here wondering whether there's a way to prove or disprove refraction.

So the ultimate problem is that, when you've already made up your mind almost dogmatically about something, then you selectively interpret and filter out evidence that seems to oppose your viewpoint.

So let me do the same analysis on this one (on the other side).  This assumes the princple that the "Horizon being always at eye level is only possible on a flat plane."

#1) is it provable from these pictures that the horizon is actually at eye level?  These are pictures, so could the angle have been changed so that it only appeared that the horizon is at eye level?  If the horizon were below eye level, and I tilted my camera slightly downward, would that make it seem as if the horizon is at eye level even if it's a bit lower?

#2) is it really true that the horizon always being at eye level is only possible on a flat plane?

Well, I believe #2 to be correct based on some other videos that I have seen on the subject ... which seem to demonstrate this visual phenomenon.  But it's not proven just by this picture alone.  And then there's a question of how the pictures were taken or produced.  So this is presented as proof, but it does not prove anything on its own.

I hope your tendencies remain consistent with the Dogmatic Flatearthists on this site. In fact, one would be hard pressed to find someone here who isn't dogmatic about it, who favors flat-earth. I can't help but hold animosity towards the FE position merely because of the words written here by the "Catholics" who push it. I am usually inclined to be open to counter-mainstream arguments and "theories" and that's the reason I didn't dismiss FE right away. It wasn't until I saw the people here pushing it and how they were doing it, that I realized this has got to have diabolical roots.

You'll need to prove the so-called diabolical roots of the flat earth position.  Saying it automatically makes you a dogmatic globalist.  Prove your position.  We've provided page upon page of proofs, including:
The idiots that run our world also insist earth is a globe (like you do),
NASA, RASA and other Freemasonic space agencies that control science are proven liars and still insist we went to the moon
The myriad of ways the world suffers from the condemned Copernican Revolution of the last 500 years
That heliocentrism, the science of the globe, was condemned by the Church,
That Fathers of the Church who taught flat geocentric earth provide a beautiful relationship of earth to the tabernacle and liturgy,
The reasonableness of flat earth with regards to what is level and what is curve
Scripture quotes on the form of the earth only support flat earth
The impossible logistics of the globe  
Ancient cartography of Catholic mapmakers
Empirical math and scientific proofs,
The warning from Scripture and from the Saints about science falsely so-called
The relationship of science falsely so-called to the imminent collapse and chastisement of the world  
All this has been covered, and more. Whether you believe it or not, it is a cohesive and well founded argument expounded upon as a whole in these threads. The information is verifiable and remains a tower against whatever spit wads you lob at it.  Not one of you globalists has offered anything but sporadic contentions, parroting anti-Catholic sentiments of the globalists. You offer a piece meal defense against a juggernaut of spiritual and physical truths. 100% of the time your claims are "no, it can't be that way" or "you're stupid", rather than explaining how you shamelessly promote what the nєω ωσrℓ∂ σr∂єr globalists do. You literally sneer at information we've provided, but most importantly, you do it without consideration. A reasonable person digests information before he spouts, asks questions, takes the time to get to the bottom of what is being said.  That NEVER happens with dyed in the wool globers, proving that indoctrination warps minds.  Modernism has been described as a system of thinking that enables one to reconcile within himself two opposing beliefs.  Globalism is the poster child for modernism and remains a scientific relativism that is undermining Christendom.            

I do not have a lot of God-given talents, but I am very skilled at shredding bad arguments to pieces.  I can see logical flaws miles away and rip them apart in seconds.  Other than that, I have a very poor memory, and not very many talents in other areas.