# Debunked: "Celestial navigation is based on elevation angles from a Flat Earth"

#### Iskweb

##### New Member
Some youtubers claim that celestial navigation is based on elevation angles from a Flat Earth. For example:
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Not only is celestial navigation allegedly based on a Flat Earth, it works so well that it amounts to proof that the earth is flat!

There are various ways to check the validity of this claim. One option is to read a book about the subject, and see what kind for formulas are used to do the calculations. A nice resource that I like to use is
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The website hosts many PDF files with formulas and explanations.

Understanding the maths behind all of this is challenging. My idea for this debunk is to use data from multiple screenshots from my Nautical Almanac App (link: https://play.google.com/store/apps/details?id=com.skrypkin.nauticalalmanac) to show that the data from my app contradicts the claim. Using a combination of times and dates and positions that I found, only simple arithmetic operations are needed to do a sanity check on the output of the program.

I will use 2 main premises to analyze the numbers from the app:
• The distance to a Geographical Position (GP) is the Zenith Distance multiplied by 60 nm. The zenith distance will be calculated as 90° – Height. Note that ‘Height’ means ‘Elevation angle’ in this context.
• If the difference in Azimuths is 180° then the distance between 2 GP’s can be calculated by adding the two distances. This is trivial, and follows from the Law of Cosines.
The first premise is commonly accepted by flat earthers, and the group I linked to will happily use it in their imagery, while celebrating their knowledge of this formula. This is one of the images they use:

And this is a modified version of the image, which illustrates the type of calculation I will use:

I took 12 screenshots from my Nautical Almanac app. They involve 4 Geographical Positions (GP's):

My app won’t show the Latitude of Longitude of the substellar points, but it will show the Greenwich Hour Angle (GHA) and Declination, from which the GP can be derived. The relationship between them two should be obvious from the table. For point D, because the GHA is greater than 180°, the longitude East is calculated by 360° - 234° 45.92' = 125° 14.08'.

Each of the 4 GP’s is associated with a moment in time when a star is directly above that position:

In celestial navigation, it is common to calculate the Height (=elevation angle) and Azimuth of a star from a certain Assumed Position and time. I have chosen positions halfway in between each combination of GP’s:

Using Assumed Position 1 and the times for A (left) and B (right):

The difference in azimuths is 180° - 0° = 180°.
The distance to both A and B is (90° - 35° 18.78') x 60 = 3281.22 nm
This gives AB = 3281.22 + 3281.22 = 6562.44 nm

Using Assumed Position 2 and the times for A (left) and C (right):

The difference in azimuths is 222° 15.06' – 42° 15.06' = 180°
The distance to both A and C is (90° - 35° 14.08') x 60 = 3285.92 nm
This gives AC = 3285.92 + 3285.92 = 6571.84 nm

Using Assumed Position 3 and the times for A (left) and D (right):

The difference in azimuths is 317° 44.90' – 137° 44.90' = 180°
The distance to both A and D is (90° - 35° 14.13’) x 60 = 3285.87 nm
This gives AD = 3285.87 + 3285.87 = 6571.74 nm

Using Assumed Position 4 and the times for B (left) and C (right):

The difference in azimuths is 312° 7.53' – 132° 7.53' = 180°
The distance to both B and C is (90° - 35° 16.41') x 60 = 3283.59 nm
This gives BC = 3283.59 + 3283.59 = 6567.18 nm

Using Assumed Position 5 and the times for B (left) and D (right):

The difference in azimuths is 227° 52.50' – 47° 52.50' = 180°
The distance to B is (90° - 35° 16.45') x 60 = 3283.55 nm
The distance to D is (90° - 35° 16.46') x 60 = 3283.54 nm
This gives BD = 3283.55 + 3283.54 = 6567.09 nm

Using Assumed Position 6 and the times for C and D:

The difference in azimuths is 270° - 90° = 180°
The distance to C is (90° - 35° 15.33') x 60 = 3284.67 nm
The distance to D is (90° - 35° 15.32') x 60 = 3284.68 nm
This gives CD = 3284.67 + 3284.68 = 6569.35 nm

In summary, we found the following distances:

AB = 6562
AC = 6572
BC = 6567
BD = 6567
CD = 6569

If we try to make a map of the flat earth that this is supposedly based on, with accurate distances, once we have placed A, B and C on it, we can’t find a place for point D such that the distances to the other 3 points are correct:

The 6 distances between these 4 points, being impossible on a flat surface, contradict the claim that a Flat Earth is being used as the basis for celestial navigation.

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Understanding the maths behind all of this is challenging.
Most people find this challenging, and Nathan Oakley is among them. That's why his argument isn't anything as complex as that. He has reiterated many times, in livestreams and shorts edited from them, that sextants measure elevation from a "flat baseline", which purportedly proves Earth is flat.

There are two obvious counterarguments:
1) tangents on circles and spheres exist
2) you can do celestial navigation with zenith angles
Either of these suffices.

A zenith angle is the opposite of an elevation, where straight overhead=0⁰ zenith angle, and 40⁰ zenith angle=60⁰ elevation.

I will use 2 main premises to analyze the numbers from the app:
• The distance to a Geographical Position (GP) is the Zenith Distance multiplied by 60 nm. The zenith distance will be calculated as 90° – Height. Note that ‘Height’ means ‘Elevation angle’ in this context.
Article:
For navigation by celestial means when on the surface of the earth for any given instant in time a celestial body is located directly over a single point on the Earth's surface. The latitude and longitude of that point is known as the celestial body's geographic position (GP), the location of which can be determined from tables in the nautical or air almanac for that year.

What we need for a triangulation is the distances to 3 known points. Below, I'll explain how to get the distance to one GP; repeat 3 times and we're done (in principle).

Since the light from astronomical objects comes from so far away, it's essentially parallel everywhere on Earth, and that gives us the GP distance from the zenith angle.

In my geogebra diagram, the distance between the observer at the "top" of the circle and the GP is proportional to the size of the angle at the center of the Earth:
45⁰×60nm = 2700 nm (nautical miles) = 5000 km = 3107 statute miles.

The hard mathematics is figuring the ground points from the astronomical tables, and calculating where you are once you have the 3 points and distances. (If you had 3 GPs and distances on Flat Earth, you'd do similar maths to find your position.)

However, there is no way on Flat Earth to get a distance from an angle observation like I show above, because on Flat Earth, either the light comes in parallel (then it's at the same angle everywhere), or it doesn't and then you need the height of the astronomical object (star), but the astronomical tables don't give you that (it's impossible).

So, not only does Nathan Oakley's objection evaporate completely when you use zenith angles, looking more closely also reveals that celestial navigation (as practiced successfully for centuries of seafaring) can't work on a flat Earth.

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He has reiterated many times, in livestreams and shorts edited from them, that sextants measure elevation from a "flat baseline", which purportedly proves Earth is flat.
I've wondered if that would mean you can't measure angles when sat on a unicycle, or resting on any type of wheel.

Kind of easy to show that angles measured when not on a wheel are the same as measured when on a wheel. But I do like the idea of an angle measuring device going all whappy if it was on a wheel.

But to be honest I dunno if that's the crux of the argument.

This is a remarkably well-chosen flat earther argument, though. To explain what they claim they are proving, they use enough math for the average potential victim's eyes to glaze over but hopefully produce an impression of the conman knowing what he is talking about because look at all that math. To come along later and then show why they are wrong requires MORE math resulting in more glazed eyes, at which point the victim is left with "well, I guess I don't really understand it, but it's easier to continue believing what the first guy told me than to have to admit I was fooled."

Celestial navigation is really interesting me cos it strikes me that it shares some common ground between flat earth and globe earth. That being a dome/the celestial sphere. Celestial navigation works on a dome that (some) flat earthers assume we are living under. This "sharing" has to be significant somewhere along the line. I'm wondering if/where (I'm currently thinking it's somewhere in the infinity of the celestial sphere but that's by the by to the reason for this post).

Now, a bit of background first. I cannot navigate celestially. I don't know the ins and outs. I'd like to think I have the intelligence to understand it if I sat down and learnt it (although I'll admit to reading some stuff but kinda start to glaze over with meridians, zeniths and what not. I quickly glaze over at the OP of this thread). But ultimately I don't need to learn it. I trust it works and is used as stated. I'll never need to navigate celestially. The incentive to learn is lacking. Please note I'm not asking anyone here to teach me celestial navigation, I'm more interested in if my (misguided) thinking on it is heading in the right direction or if I'm miles off.

That said, I get that GP (ground position) is important when it comes to celestial navigation.

I'm not too sure how to say this without coming across as a total dumbass (but hey, when learning we all enter as a dumbass), or how to articulate it correctly, but I think my uninformed mind assumed we knew what the GP of stars are. A map, a chart, I dunno, but I just thought it was something we "knew" and that gets referred to for navigating.

But then I started to think that was maybe a bit stupid. I stand outside and look up, how the heck do I know what star is exactly above me? And it moves. There'll be a different one in a while.

Again I'm gonna struggle to articulate here, but instead of it being something we've mapped, the GP is fundamentally the mathematical result of a sphere inside a sphere.

In other words, it seems to me that celestial navigation presupposes a globe. And surely that's a criminal offence in flat earth circles. I dunno. That should smart, shouldn't it?

And if that is the case then it obviously wouldn't be possible to use that math to calculate the GP of a star on a flat plane (scale being important). Although that of course is not saying that the math to calculate the GP on a flat plane is impossible. I'm sure it could be done (by someone much smarter than me) although the introduction of a dome over a flat plane really does start to introduce a lot of "problems" that flat earth has with the globe (like wouldn't there be 2 GPs? one plumb to the flat plane and another at the tangent of the dome? Tangents are kryptonite to flat earthers)

But I'm getting side tracked. Is it fair to say that the (calculating) GP of stars for celestial navigation presupposes a globe? (without getting too triggered by the definition of presuppose, I hope you see what I'm getting at).

Thanks.

You've raised a good question.

Celestial navigation is really interesting me cos it strikes me that it shares some common ground between flat earth and globe earth. That being a dome/the celestial sphere.
This a really important observation. With all of the changes in the sky, we can understand them (approximately) as the stars being pinned to the inside of a celestial sphere; and the moon and sun moving around in a circle(ish) path once every 27/365 days. This means you can make a "sky globe" with the stars on it. The ancient Greeks left us a statue that had one.

And if you look at the night sky, wherever you are, you can see that sphere rotating. With no distortion. See e.g. https://www.metabunk.org/threads/be...t-show-the-earths-curvature.11456/post-244218 .

If you're ever in a dome building, check to see that the proportions of the dome change, depending on where you are on the floor, due to perspective. The star sky never changes, the constellations never change apparent shape, no matter where you are on Earth.

The celestial sphere has two poles (centers of rotation) and an equator, like Earth.

But then I started to think that was maybe a bit stupid. I stand outside and look up, how the heck do I know what star is exactly above me? And it moves. There'll be a different one in a while.
"Zenith" is another word for "exactly above". You could use a plumb line to figure out the approximate direction. You could use a spirit level to establish horizontals of 0⁰ elevation, and then look at 90⁰ elevation. (You could craft a simple astrolabe.)

Measure the elevation of the celestial pole that you can see to find your latitude. They're the same. It follows that the celestial North pole is directly above Earth's North pole (which is fortunately near the magnetic North pole).

Flat Earthers deny there's a South Pole on Earth. They have a hard time acknowledging the celestial South Pole, which you can see in the night sky in the southern hemisphere as the stars rotate.

How high up is the star Polaris, near the celestial North pole? Globers say, pretty far away. FEers put it at a few 1000 km, but can't work out exactly how high.

Again I'm gonna struggle to articulate here, but instead of it being something we've mapped, the GP is fundamentally the mathematical result of a sphere inside a sphere.
As it's used in celestial navigation, yes. The fact that these sphere-in-sphere maths align with where the stars actually are proves the globe. There is no "flat Earth" method to predict that.

If you have a sky globe, to find the star above you take the ring that fits your latitude, and then rotate it to match the date and time. (Clocks are very important for celestial navigation. The ball dropping in Times Square in NYC at New Year's was originally used in port towns to synchronize the ships' clocks to the accurate time at noon.) Then look straight up, at your zenith, and it's there.

For that to work, i.e. for the latitudes to match, Earth must have the same curvature as the sky globe. Earth must be a globe. As soon as you verify that the sky globe works, you know that.

Some people think that Erathosthenes proved that Earth is round with his "shadow in a well" experiment. But ancient Greece had the sky globe, so Erathosthenes already knew that; all he wanted was to figure out how big Earth is.

The only way the Flat Earth sky would work was if there was a flat map of it. But as with Earth's surface, there isn't one (that works).

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P.S.:
the introduction of a dome over a flat plane really does start to introduce a lot of "problems" that flat earth has with the globe (like wouldn't there be 2 GPs? one plumb to the flat plane and another at the tangent of the dome? Tangents are kryptonite to flat earthers)
That's a smart observation, too.

If the dome is a hemisphere, then "straight down" as in "perpendicular to the tangent" always points at the terrestrial North pole.

You can only have the "tangent down" and the "plumb line down" coincide on Flat Earth if the sky is flat, too.

Like I said in my previous post, the only way the Flat Earth sky would work was if there was a flat map of it—but there isn't.

I think you may have answered it (I think you're saying yes) but, as I said, I glaze over at a lot of the minutiae of this topic (and I wouldn't be surprised that's the case for the majority of flat earthers as well. Please note I'm not saying I'm a flat earther there).

I did state I'm not asking to be taught celestial navigation here.

I don't think you have to totally understand all that minutiae either to see how celestial navigation is sphere inside a sphere instead of dome above flat plane, which is where I'm kinda going with this.

With all the cries of presupposing a globe from flat earthers it just struck me as it'd be ironic if they stole something that presupposes a globe.

One thing you could possibly answer, and it relates to all the mention in your post of it not working on a flat earth....

In my googling about GP I happened across this site...

http://homework.uoregon.edu/pub/emj/121/lectures/skycoords.html

... which in talking about Altidude-Azimuth Coordinates states...

This model assumes that the earth is flat (with respect to the observer)
... and so to someone as uninformed as me I wonder if that makes me wrong about the whole presupposing a globe thing? Or how you can say it doesn't work on a flat earth?

Is it possible without getting bogged down in the specifics (that I glaze over on) what they mean by that?

Please note I'm not trying a "gotcha" here. I know why airplane flight stuff often says the same and doesn't mean we live on a flat earth.

Hopefully you can see why something like that said on that site has me questioning my idea on it presupposing a globe (which I do think is correct).

Like I said in my previous post, the only way the Flat Earth sky would work was if there was a flat map of it—but there isn't.
Flat maps of curved objects.

It's ironic.

I think you may have answered it (I think you're saying yes)
That's correct.
I did state I'm not asking to be taught celestial navigation here.
I didn't (can't do it myself). I stopped at the point where time gets involved because that's where the maths starts.
I don't think you have to totally understand all that minutiae either to see how celestial navigation is sphere inside a sphere instead of dome above flat plane, which is where I'm kinda going with this.
If you can put it more simply than I could, that'd be great.
With all the cries of presupposing a globe from flat earthers it just struck me as it'd be ironic if they stole something that presupposes a globe.
It is ironic.
One thing you could possibly answer, and it relates to all the mention in your post of it not working on a flat earth....

In my googling about GP I happened across this site...

http://homework.uoregon.edu/pub/emj/121/lectures/skycoords.html

... which in talking about Altidude-Azimuth Coordinates states...
This model assumes that the earth is flat (with respect to the observer)
Content from External Source
... and so to someone as uninformed as me I wonder if that makes me wrong about the whole presupposing a globe thing? Or how you can say it doesn't work on a flat earth?

Is it possible without getting bogged down in the specifics (that I glaze over on) what they mean by that?
They're just saying that so they can use the word "horizon" to refer to the 0⁰ elevation line/plane. On the globe, the actual horizon typically has a slightly negative elevation. But it's more intuitive to use than the zenith, because most people have looked more at the horizon than at the zenith in their lives. And what they're explaining there doesn't depend on the shape of the Earth, only on the shape of the sky.

They make up for it later when they say,
It should be apparent that this system is based on the observers location on Earth.
Content from External Source
Hopefully you can see why something like that said on that site has me questioning my idea on it presupposing a globe (which I do think is correct).
Celestial navigation, with a sextant, as practiced successfully on oceangoing ships for centuries, only works on the globe. The maths underlying it depends on the "sphere in a sphere" model and the planetary motions as described by Kepler (matching Newton's gravity).

And what they're explaining there doesn't depend on the shape of the Earth, only on the shape of the sky.
And would that be because what they are explaining in that portion isn't about finding a specific location on the assumed flat earth? Just how terms used are defined?

Apologies for having to dumb it down so much.

And would that be because what they are explaining in that portion isn't about finding a specific location on the assumed flat earth? Just how terms used are defined?
Yes, exactly.
Apologies for having to dumb it down so much.
I see it as supplying you with information you don't yet have. That's always a bit of a stab in the dark, as we both don't know exactly what it is that you don't know yet.

You've clearly put some thought into this and are articulating your questions well, so it's fun—no need to apologize.

Regarding the infiniteness of the celestial sphere...

Put anything in the centre of it and all the angles are all the same?

While the concept of GP in celestial navigation presupposes a sphere inside a sphere it doesn't presuppose the size of the inner sphere?

No avoiding the plumbs are/aren't parallel aspect of this but it seems interesting that GP appears to me to be working on the angle that plumbs of off parallel to each other.

It's not hard to see how the angles between plumb are always the same regardless of the size of the inner sphere, but I dunno, it seems kinda profound that the angle measured from the surface would always been the same. Assuming I'm correct in my thinking?

i.e. an angle measured to a star on the celestial sphere from the surface of an atom would be the same as an angle measured from the surface of the sun (assuming the same orientation on the sun/atom and emphasis on surface).

That we personally can effect the angle just by moving up or down on a scale that seems insignificant to the scale we're working on (globe, infinite) it kinda feels wrong that the angle doesn't change regardless of that scale.

Or am I miles off? (navigation pun, teehee).

The angles work the same if you move the same number of degrees on a sphere of any size. However those degrees will translate to different distances on spheres of different sizes. The distance you walk to traverse 1 degree of latitude on Earth will go around a bowling ball numerous times, and would go less than 1 degree on a hypothetical super-Earth.

So while the principles of solving for your location would be the same, how that solution would translate to actual navigation choices would be different. Celestial navigation in the theoretical sense works on a billiards ball as well as it would on the outer hull of a Dyson Sphere, but in the practical sense, as used on Earth, it presupposes not merely a sphere but a sphere reasonably close to the known diameter of Earth.

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The angles work the same if you move the same number of degrees on a sphere of any size. However those degrees will translate to different distances on spheres of different sizes.
Yeah, I do get that but I'm not there yet.

I'm just looking at the concept of GP as it is proposed for celestial navigation. From my uninformed position (navigation pun, teehee).

I'm just circles and fixed surface and sky at the moment. I'll happily admit I have no freakin' idea yet what use knowing the angle that a star's plumb is to mine (I could be totally wrong as to that being what GP is for all I know) is to finding my own position (please note that's not a request to have that explained here and now. I promise I'll glaze over any attempt and anyone would just be wasting their time).

I'm just enjoying the wonder of circles at the moment and looking for validation of my reasoning and discoveries from those more informed than myself so I know I'm heading in the right direction or not (yep, navigation pun, teehee).

While GP not presupposing the size of the sphere (in regards to angles) might be obvious to some it came as quite a surprise to me. Assuming I'm correct in that claim. Which I still don't know if I am or not.

No avoiding the plumbs are/aren't parallel aspect of this but it seems interesting that GP appears to me to be working on the angle that plumbs of off parallel to each other.

It's not hard to see how the angles between plumb are always the same regardless of the size of the inner sphere, but I dunno, it seems kinda profound that the angle measured from the surface would always been the same. Assuming I'm correct in my thinking?
I think you are referring to this:

Since the light from astronomical objects comes from so far away, it's essentially parallel everywhere on Earth, and that gives us the GP distance from the zenith angle.
Note I didn't put a scale on that drawing! It's not necessary.
That we personally can effect the angle just by moving up or down on a scale that seems insignificant to the scale we're working on (globe, infinite) it kinda feels wrong that the angle doesn't change regardless of that scale.
You're using a not so useful frame for your thoughts. In maths, the scale is not so important. If you draw two intersecting lines on a piece of paper, then the angle is going to stay the same no matter whether you zoom in or out.

The part where scale comes in is this: we know that a star sends out light in all directions there are. So why is the light from the star that arrives here parallel? Shouldn't it be different directions?

That's where scale is important. The sun is 150 million km away, Earth has 12,742 km diameter. This means that the sun is ~10000 diameters away. This ratio of 1:10000 corresponds to an angle of 0.006⁰. When a degree is 60 nm, 0.006⁰ are ~ 500m.

The nearest star is Alpha Centauri, at a distance of 40,000,000,000,000 km the accuracy of the angle is a million times better, and the distance error would be less than 1mm (1/32").

So what matters here is that the scale of our position variations is so small compared to the distance of the stars that the error we're introducing by treating the light as parallel is very very small, too.

The intuitive version of this:
if you and a buddy look at a candle on the table you're sitting at, the direction each of you sees it will be different; but if you look at a didtant lighthouse, or the moon, the direction will be the same because these lights are far away.

I could just give a thumbs up to @Mendel's post, but I want to support it more strongly than that. (Including his quoted previous post.) This is well-established maths, it could be drawn out in to tedious numerics, but there's very little point - if this were wrong everything else would fall apart. Even Euklides would approve (but maybe we're just in cahoots with Euklides?! (Have I been taken over by the ghost of Lobachevsky?!?!?!)).

This is well-established maths, it could be drawn out in to tedious numerics, but there's very little point - if this were wrong everything else would fall apart.
Surprisingly, this is true.
The "blow my mind" version of this goes, all circles are the same (except for size and location). Two triangles can [be] different, but two circles can't be different.
And as Euclid's third postulate, this has been a foundation of geometry for over two millennia.

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Surprisingly, this is true.
The "blow my mind" version of this goes, all circles are the same (except for size and location). Two triangles can different, but two circles can't be different.
And as Euclid's third postulate, this has been a foundation of geometry for over two millennia.
Postulate, not axiom ;-) +/0/- who knows?

Surprisingly, this is true.
The "blow my mind" version of this goes, all circles are the same (except for size and location). Two triangles can different, but two circles can't be different.
This is true for other geometric shapes too, like squares and parabolas. Circles aren’t unique in this respect.

This is true for other geometric shapes too, like squares and parabolas. Circles aren’t unique in this respect.
squares and parabolas are oriented, circles are not (in 2D). A corner of a square can point north (or not), try that with a circle!

squares and parabolas are oriented, circles are not (in 2D). A corner of a square can point north (or not), try that with a circle!
Fair enough. I was speaking more about the shape itself, which is invariant under translation and scale. You hadn’t specified a coordinate system.

A circle is indeed unique in having an infinite number of symmetry axes.

If the dome is a hemisphere, then "straight down" as in "perpendicular to the tangent" always points at the terrestrial North pole.

You can only have the "tangent down" and the "plumb line down" coincide on Flat Earth if the sky is flat, too.

Like I said in my previous post, the only way the Flat Earth sky would work was if there was a flat map of it—but there isn't.
I do like the idea that the dome could provide a second angle that flat earth requires.

Flat earth's GPs are all parallel so what's the need for this other angle that sphere inside sphere appears to be working on? What if it's the angle to the north pole (or based on the curvature of the dome if the dome isn't a perfect hemisphere)? That's how "they" (that did all the math) are hiding the flat earth in celestial navigation maths.

The very notion that people who understand it all could be tricked in that manner is utterly preposterous to myself but I doubt it's too much a stretch for flat earthers (if they don't understand it then no one does). Although it does seem like to do celestial navigation you don't need a deep understanding of all the principals it works on. People have done the work to make it easier and accessible to more people. So you could argue that it's hidden in there somewhere. But again, that there's no one with better understanding of the principals than me is ridiculous.

BY FAR the most interesting thing about this is that I personally lack the knowledge to understand an explanation of why this particular idea is wrong. I dunno but that feels like a heck of an insight into flat earth. And also the fact that even though I've told you I wouldn't understand an explanation I bet there's people who are dying to give it anyway. I dunno, but proposing something I'm unable to understand if it's right or wrong seems so alien and ridiculous to myself but I think it's actually part of what flat earth is built upon.

Ignoring how celestial navigation DOES actually work, I'm intrigued if navigation of a flat disk under a dome could even work on the idea I've suggested here? Again, I lack the knowledge to know if it would or wouldn't. I like to think it would and the curve of the dome could be worked out, although I suspect my method would need to presuppose the curve of the dome in order to work. I doubt there's anyone with the required level of knowledge to care and find out though.

Apologies for the lack of navigation puns.

Flat earth's GPs are all parallel so what's the need for this other angle that sphere inside sphere appears to be working on? What if it's the angle to the north pole (or based on the curvature of the dome if the dome isn't a perfect hemisphere)? That's how "they" (that did all the math) are hiding the flat earth in celestial navigation maths.
The problem is that the angles don't work out.
Although it does seem like to do celestial navigation you don't need a deep understanding of all the principals it works on.
Working out your latitude is dead easy on the Northern hemisphere, make yourself a simple astrolabe/clinometer and observe the elevation (apparent altitude) of Polaris (if you want to be more exact, the celestial north pole), and you're done. (Longitude is the hard part. Or if you have to it when Polaris is obscured. Or if you need precision because your ship might run aground otherwise.)

For the ancients, the latitudes of Athens, Alexandria, and Assuan are 38⁰, 32⁰, and 24⁰, so there's quite a bit of observable "dome tilt" across the Mediterranean Sea and along the Nile; the Greeks and Egyptians could hardly miss it.

But the fact that 1⁰ of tilt = 1⁰ of latitude works out to approximately 60 nautical miles north-south distance everywhere on Earth proves we're on a sphere. You can just draw this geometrically for both models and compare (see above).

Clinometer (measures zenith angle):

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The problem is that the angles don't work out.
Good point.

I think where my brain fails is it's inability to comprehend that right hand diagram but instead always sees it as...

I think my "Regarding the infiniteness of the celestial sphere..." post was trying to reconcile this mental picture into the right hand diagram but going about it in the wrong way.

It feels like a paradigm shift that my brain just rejects and will never accept. Until it can then I'm all lost at sea (sorry, couldn't help myself).

I think where my brain fails is it's inability to comprehend that right hand diagram but instead always sees it as...

I think my "Regarding the infiniteness of the celestial sphere..." post was trying to reconcile this mental picture into the right hand diagram but going about it in the wrong way.
There three ways to go about it, that I can think of.

The first way is to make the circle smaller (zoom out) and then move the star further away. As you do that, the lines get more and more parallel. If you have a vector drawing program, you can try that out yourself; if you don't, I could probably set something interactive up in geogebra.
They key notion here is that the rays are never quite parallel, but they're so close to it that the difference is small enough to not matter. (Kinda like Earth looks flat until you zoom out.)

The second way is to go 3D and imagine the star on the horizon, and the rays lying flat, because then perspective makes it look like the star is far away and the lines look parallel. For that, it might be useful to add a horizon and some perspective objects to the diagram.

The third way is to draw the star and Earth the same size (the star is actually many times bigger), and then apply the first way.

You're correct, though, that it's intuitively easier to think of the stars as small and near. (The Greeks probably did.) But even so, the change in tilt of the celestial sphere when you move around the Earth is observable for anyone with a sky globe, and only works if you're on a sphere yourself.

Good point.

I think where my brain fails is it's inability to comprehend that right hand diagram but instead always sees it as...

I think my "Regarding the infiniteness of the celestial sphere..." post was trying to reconcile this mental picture into the right hand diagram but going about it in the wrong way.

It feels like a paradigm shift that my brain just rejects and will never accept. Until it can then I'm all lost at sea (sorry, couldn't help myself).
This drawing is extremely out of scale for a star. Even a geostationary satellite is farther out than what you've drawn. A geostationary satellite is about 6.5 Earth radii above the Earth, and the closest star (besides the Sun) is over 6 billion Earth radii away.

This drawing is extremely out of scale for a star.
That's the bloody point of the diagram!!!!!!

Sheesh.

Closer (but still way off) to scale, it maybe gets a little clearer. Or maybe not, depending on what various brains chose to do with the image.

Am I correct in imagining that an infinitely far away night sky above a flat earth would be a single star?

Am I correct in imagining that an infinitely far away night sky above a flat earth would be a single star?
Is it infinitely wide? (I'll admit to having conceptual issues when comparing infinities, it is very possible that some mathematician will explain that it would make no difference... )

Is it infinitely wide? (I'll admit to having conceptual issues when comparing infinities, it is very possible that some mathematician will explain that it would make no difference... )
You'd have to ask if flat-earthers believe there are stars that we can't see because they're "beyond the edge". The answer is undoubtedly going to be both yes and no, because each seems to have his own concept. There is no coherent and agreed-upon definition of what a flat earth would look like, let alone the sky.

There is no coherent and agreed-upon definition of what a flat earth would look like, let alone the sky.
Absolutely, it is all ad hoc to meet the current point of argument, but often not consistent from moment to menent in discussion with an individual flat Earth believer. I've reached the point of just saying "Come back with a flat world map with a scale and we can talk." So far, nobody has. For obvious reason!

Am I correct in imagining that an infinitely far away night sky above a flat earth would be a single star?
I don't understand what you're imagining here.

If the stars were very very far ("infinitely") on Flat Earth, the sky would look exactly the same everywhere on Earth. You could imagine a tiny postage stamp-sized flat Earth inside the "sky sphere", like e.g. everyone in the same town sees essentially the same sky.

This is not a problem conceptually, it's a problem empirically because we know the sky changes once you travel long distances. (1⁰ of tilt per 60 nautical miles.)

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You could imagine a tiny postage stamp-sized flat Earth inside the "sky sphere", like e.g. everyone in the same town sees essentially the same sky.
OK, I can see I was wrong.

I'm intrigued by your "same town" comment. Wouldn't it be the whole postage stamp sees the same sky (assuming everything's stationary)?

OK, I can see I was wrong.

I'm intrigued by your "same town" comment. Wouldn't it be the whole postage stamp sees the same sky (assuming everything's stationary)?
Yes. Kinda both.

If the town is less than 6 nm (11 km) across, the variation in sky tilt is less than 0.1⁰, or ⅕ of the moon. That's "essentially the same" for me, I'm fairly certain I wouldn't notice.
On FE, there'd be no visible tilt no matter how far you go, obviously.

Yes. Kinda both.

If the town is less than 6 nm (11 km) across, the variation in sky tilt is less than 0.1⁰, or ⅕ of the moon. That's "essentially the same" for me, I'm fairly certain I wouldn't notice.
On FE, there'd be no visible tilt no matter how far you go, obviously.
There's one circumstance in which you'd notice: an eclipse. A couple of years ago there was a solar eclipse, in which my daughter hosted a watch party because she lived on the side of town where the eclipse was total, while those a very short distance to the east only saw a partial eclipse. I wonder how the FE crowd explains that.

Yes. Kinda both.

If the town is less than 6 nm (11 km) across, the variation in sky tilt is less than 0.1⁰, or ⅕ of the moon. That's "essentially the same" for me, I'm fairly certain I wouldn't notice.
On FE, there'd be no visible tilt no matter how far you go, obviously.
You seem to have come up with a number for "very very far", as that's one of the inputs necessary for getting a number out.
As your parenthetical "infinitely" indicates, you're justified in just taking the limit as the number you chose tends to infinity, and that leads to a tilt of 0⁰ in the limit (which is well behaved). You were right at the outset, the adding of numbers has fuzzied things.
"All finite things are but nothing to the infinite" - Epifatphilus, famous non-existent non-greek non-philosopher.

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