Okay, I'm the one who started this. This will be my final input. But I'd like to make a few last points/clarifications:

1) I guess in the beginning I should have defined terms. What I mean my "celestial navigation" is the final product of years of smart people working on this problem of using a sextant, a chronometer, and carefully developed tables or computers/calculators to find your position at sea. The math required to do so is spherical trigonometry. This is a fact. Spherical trig is used because you are finding your position on a sphere. So, my definition of "celestial navigation" is the common system in worldwide use before the advent of electronic navigation systems like GPS, LORAN, DECCA, etc.

2) It is true that people have been using other kinds of celestial navigation for millennia. The indigenous navigators of the South Pacific are particularly famous. (Read the book about them: "We, the Navigators" by David Lewis - fascinating!) Even in more modern ages there is a "lunar method" that does not require a chronometer. The famous sea captain, Joshua Slocum, who was the first person to sail single-handed around the world in 1895-98 reportedly used that method, but I understand it is extraordinarily complicated. However Slocum spent years before that famous voyage as a regular sea captain, so he had the experience & knowledge.

3) The example I gave of a large spherical triangle based on the pole and the equator was just that, an example. One easily imagined in someone's mind. In real life celestial navigation involves solving many different spherical triangles, some big, some small. There are an infinite number of possible spherical triangles on the surface of a sphere, just like there are an infinite number of potential flat triangles on a plane. I was just giving an example. Don't get hung up on that.

4) It's also true that navigators use flat maps (charts) at sea, not a globe. BUT, BUT, BUT that's because flat maps are much more convenient, AND over the scale of the typical marine chart the differences between a flat Mercator projection chart and the real spherical world are trivial. But that is for the same reason that when you are standing on shore looking at the horizon - it looks flat. It's NOT FLAT! But the curvature is so subtle that the human eye and brain cannot detect it. Similarly over the distances covered by a typical chart "correcting" for the fact that this is a flat representation of a small portion of a sphere is just unneeded. BUT for crossing entire oceans, then it matters. You then use a great circle course, which when plotted on a standard Mercator projection chart ends up looking like a "longer" curved line, but in the real world is the shortest distance. The ONLY great circle routes when plotted on a Mercator chart are still straight lines are at the Equator headed due east or west or when traveling absolutely due north/south along a line of longitude. All other great circle routes on a standard chart will look like a curved line. In an earlier post I mentioned Charles Lindbergh. Before his historic New York to Paris flight, he went to the main New York City public library and stretched a string on a very large globe they had there (I understand it was 4-5 feet in diameter) between New York and Paris. Since he could not take a globe with him in his airplane, he took careful notes as to where he needed to be for each stage of his flight to follow a great circle course. And in almost every book I've read about Lindbergh and his flight, there is a chart of his course across the Atlantic plotted on a Mercator projection chart of the Atlantic ocean. The course looks like a curved line. In reality, up in the air, it was not - it was a straight-as-he-could-do-it straight Great Circle line to Paris.

4) And again, all those crusty old sea captains who had to do celestial never fessed up? And none of them ever found the "ice wall" circling the flat earth? And since the southern oceans were at one point heavily sailed by whalers (thankfully that's over) you think one of them would have noticed an endless wall of ice?

5) And I need at least mention another inconvenient fact about celestial that I haven't yet mentioned. Celestial is also based heavily on a bunch of smart astronomers figuring out precisely where the "geographic point" is for all the celestial bodies used for navigation are at any given time. The celestial bodies are: sun, moon, major planets, and about 20 of the brightest stars. The principle is using the angle measured by a sextant and a chronometer to calculate where you are compared to the geographic point of the celestial object you are using at that exact point in time. (Given the rotation of the Earth, for every 4 seconds you are off on your time measurement your calculated position is off by a mile of longitude.) The geographic point is where the celestial object would be directly over your head. If the object is low on the horizon from the navigator's perspective the geographic point could be thousands of miles away. That's not a problem, but then the use of spherical trig is mandatory or else the error involved would make the effort pointless. Plus, that entire process of calculating and then incorporating the geographic point for all those celestial bodies into the tables or calculators/computers used is also based on the earth being a globe. None of those astronomers are flat-earthers. So geographic points are yet another critical part of celestial navigation that relies on the Earth being a sphere.

6) To the FE people who have replied, it strikes me that none have you have carefully thought about what I and others have said about celestial navigation. Your response seems to be the intellectual equivalent of putting your fingers in your ears and yelling "Na-Na-Na-Na" when someone is telling you something you don't want to hear.

7) And no FE responder has explained why the distances required for an around-the-world sailing race would be 2-3 times longer based on the common FE model than they really are. And, yes, the sailors would notice that sort of thing.

Finally - Celestial navigation works. It's worked for centuries. Long before NASA and at least since the 1760s by every sea captain from every seafaring nation, regardless of politics, religion. The math required is spherical trigonometry. It's spherical trigonometry because the earth is a sphere. No one would bother using the complexity of spherical trig if it wasn't needed. Be intellectually honest and get over it. I'm going sailing.